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Mechanical properties of steel 16: solid solution strengthening
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Title  Mechanical properties of steel 16: solid solution strengthening 
Title of Series  Mechanical properties of steel 
Part Number  16 
Number of Parts  24 
Author 
Cooman, Bruno C. de

License 
CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  10.5446/18321 
Publisher  University of Cambridge 
Release Date  2013 
Language  English 
Content Metadata
Subject Area  Engineering 
Abstract  The 16th in a series of lectures given by Professor Bruno de Cooman of the Graduate Institute of Ferrous Technology, POSTECH, South Korea. Deals with the solid solution strengthening of iron an steel. 
Keywords  The Graduate Institute of Ferrous Technology (GIFT) 
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Transcript
00:04
arrange so we have the
00:19
strengthening mechanism due to solid solution in Inc and again as we are doing in this course focusing on now both are endemic Maryland and that steals the alloys of used and so sold but and we were going through the effects that solutes have 1 day when they strengthened the the steel so that the 2 main effects Over sighs of facts since an and would call the modulus effect to the size affects very simple to understand why you introduce when atoms that may be for instance much larger than the iron after yes offerings so you introduced tungsten In 1 of the farright but after it's going to generate the distortion in the lattice and power and and that will have a president will see strengthening effect 2nd or you can have it added an air an an element that did justice just a country of for instance silicon or aluminum little smaller and do little given that the latter's contraction and animals will love the use of a parameter called Delta to describe this strain this volume strain that you get around it some the other thing is when you add an element to the steel you a locally you change the elastic properties what you actually do is change the electronic properties and and the binding but you can understand this in terms of our of making this to material locally softer or harder knowing that this will also have an impact on the the strength How can you force for instance you can have our halted to the very simple examples of vacancy it's not really a malady known as the effects it has basically no the modulus it's it's modulus equals 0 right that'll have very strong influence on the deal today the strength of the material and because of this big difference spent much of this and then there are a number of the other less why is this this to move so the the decisive
03:16
factor we use a parameter Delta which is 1 overriding the ADC aid being the lattice parameter yes and so the ABC is the dependency of the change of the last forever with the concentration of the solitude and if you divide this by is basically a strain stress the modulus would do the same thing as 1 over GED GDC relative change of the models when the concentration of the source and then as I said you have other effects which may or may not have an influence we have what we call the chemical interaction and when we refer to chemical interaction we typically referred to the influence what you factor that solutes have on extended dislocation status and so on of this would be in effect that only occurs when we have stacking falls so in low stock informative but doesn't really happen in the Bay Area in um a variety of frantic steals and and the reason is that the solutes maybe the but migrating to the stacking faults and having an influence on the the motion of the dislocation and in this way these sometimes when you add enough solutes yes and you can have a phenomenon of ordering for instance if you add the silica too variety at 1st the solution will be randomly arranged and then as you add more and more of 4 of the past 10 or 12 atomic percents of silicon similar things happen with I aluminum you get awarded face silicon and an iron will force a new Crystal lying structure because the silicon is ordered and you form what it called a deal with restructure for instance for that now with the which has a chemical composition of at the 3 aside or at the 380 out that which also have in alloys is that you have and then when you form as high at the 3 or the I C 3 you have long range or all that kind of order can occur at lower concentration in which we talk about shortrange order if you wish to have patches of ordering and then there Of course ordering can only have effects when you have enough alloy Yang at present more how it does and what it basically would meet organizers of preferred local arrangements of iron atoms and solutes now that when you have a in the normal dressed all the doesn't have ordering the when a dislocation passes through the press before did before this year the dislocation you have perfect crystal and after the dislocation has passed Chrysalis perfect too there's no change however when you haven't wanted structure most of the time when you have a dislocation passes through it this ordered structure is not capped after the dislocation has passed so there is an energy difference before and after the passage of this this and this also leads to a strengthening effects In because the shortrange order this present you the passage of this was partially removes across the blight plane and you basically randomized depressed or has had and that has effect on the but stress you need to move next door the next dislocations through that same region In steals yes we have a special kind of strengthening the facts which are due to the very strong relations between interstitials interaction the strong interaction between interstitials and dislocation and we talk about carbon and nitrogen interstitials mainly and where we have aid phenomena that's called smoke or
08:37
drink ordering is very simple fact it it is due to to the the fact that carbon and nitrogen interstitials have a high diffuser fitted so at room temperature this carbon and nitrogen and it makes 1 jumped 1 jump from 1 of 2 he took place to the other opted replace yes so you can have In fact the their duel or interactions that their due to this very shortrange diffusion yes and that is what's new ordering is about is that you have dislocations passing through the lattice yes and at room temperature the carbons indivisibility of this that are indivisibility of the core of the dislocation can move from and To and energetically more favorable Place yes and thereby they but slow down they form obstacles to dislocation will talk about this in more detail because it is important what does it mean for instance you do you take and a frantic steel the carbon steel and I used you love you do you measure the stressstrain curve but you don't go all the way you stopped after a few per cent of strain yes so originally there is no I yield .period or anything you have a nice continues yielding behavior so you to form a little but few per cent 3 4 per cent you stop yes you use removed the load and after a few seconds you measure again and lo and behold you have yield .period there yes that yield .period has is that the debate the proof of this suit ordering right and we also have electronic interaction so when we consider a location in a metal yes you wouldn't think much of France by electronic affects about we have stresses yes in the vicinity of the dislocation and that this stirrups the density distribution of electrons there are now regions with different electronic densities we're not going to describe them in detail but so you can imagine different densities electronic densities and then when you add an atom yes which have as a different of ballots different balance valency and then irony L that's better the atom will interact electrically as at war with the the With this event With this new electronic distribution around the core of the dislocation so what but so electronic effected changes In electronic density yeah electronic is you might wanna like when we talk with electronic devices that there are some materials where the dislocation actually charged like if you think about ionic crystals like sodium chloride you also have dislocations there there necessarily have a charge at the end of the of the dislocations you you haven't and electric shock but in the case of the the map tells you don't really have a charge but you have the distribution of electrons is so good and so electrons move away from regions that are in compression and and rich as it were to get rid of the region's other tension and so you get a dipole a dipole an electrical dipole and a dipole can interact with the charged particle units and that would be charged particle is an element solitude you have added which has a valency different than 1 of them that of 5 OK is that an important factor no basically I 4 of the 6 facts the big Maine 1 of which we know a lot is the size of France has even if you wanted to take electronic effects into account you'd be United have lots of difficulties finding any data about case right so
14:03
let's start with the simple approach which which which is the engineering approach with those solutes strengthening as well it's an empirical approach which is assume that the strengthening is proportional to the solutes concentration in mass precise right but there is no strong theoretical support for this approach yes but there are theories which gives a linear relations between the amount of strengthening and the concentration of the solid but there are other theories also is not as no strong but that support for this approach but in practice it works and the reason why it works is and of our make this point lead run is that we don't have much experimental evidence 2 0 support any Of the available theories when it comes to solid solutions strengthening in steals so when its and the results are not good enough in a very often because because the levels of alloy Ewing's are so low but in so we knew that we assume a linear relation with and when we when we do this we see that the interstitial solutes like carbon nitrogen may have very high strengthening effects and forensic steals phosphorus silicon manganese next the ones that have an intermediate strengthening effect and then you have elements like Nicole Molly have weak effects and in certain cases of chromium can even have a softening effect that anything you know about solid solutions strengthening in the far yes it does not necessarily hold for Austin yes so manganese is a solid solution strengthen our In there are 8 yes but it actually it's softens Austin so when you have an Austin exterior and manganese a dozen gets harder In fact the many cases where manganese makes it's a softer material so the very careful OK but as a rule interstitials increase the the strength in Jakarta when the nitrogen have high strengthening effect in all Semitic steals and in silicone tight Tamiment Molly have a strong intermediate stabilizer learned strengthening effect manganese and nickel do not have a pronounced solicitors struggling effect in up
17:10
and so we have lots of data 2 to rely on foreign is Arron phosphorus graph you can see nice linear relation but with silicon it's all there is a lot more scattering the data that we can draw a straight line through it the Arab aluminum then straight line as for but which supports this empirical approach reciting while strengthening is proportional to the birth weight per cent or mass content of the underlying element here and the same thing for Our Austin of Semitic steals is an example here With an example for eclipsed the old at some aluminum you see a linear increase in distress so by then and and so you can derive from this 20 major Pascal's the mass cent of aluminum that you act now so you can use this in practice however when you do this and this is a list of the some of the solid solutions strengthening factors that I have found in the literature I so you think the thanks 1st of all but there's lots of data out there so many people have looked at this a 2nd if you look at the data it's very puzzling right take for instance the interstitial hardening effects yes carbon 5 thousand 500 44 mega Pascal per weight per cent is 1 of the measurements but 1 of them reported mentioned 2 thousand 263 is another value yes so a huge deference obviously knows all of the same nitrogen renege 5 thousand 15 years but some elements you have very little information has for long I only find 1 1 value phosphorus same thing goes 1200 500 silicone 100 40 60 32 18 2 huge differences while Saskatchewan well this of course reflects the fact that when people do these measurements yes very often they do this on stilts yes the concentration of the elements but it is never measured perfectly In this sense that difference in the case of this carbon here are we sure that a person who did the measurements checked but there was no carbon present as 7 tight all a transition carbide How sure and we know that the Carbone yes but was not segregated for instance to grain boundaries we all we don't know this information but it's very important of course so ideally yes the best measurements are the ones but they're made on single crystals and perfect binary solutions so there shouldn't be any precipitation but obviously the case of carbon it's really difficult to control because carbon is basically insoluble In fare so how can you get reliable data so it's a challenge well so when what do I do much of this is that it's an empirical approach 0 know and it's not based on theory so what do you do in practice but because it's not that these these measurements of back in know all this is a bad measurement and this is a good measurement I have you know they're all good measurements were based on you know the author measuring a different manganese content put it through a line through it and that's what that that person obtains the disbursed there's nothing wrong with it so what I do is up I collect these and I choose the median value yes I could the meat not the mean value the medium that's the value that's in the middle of the road right so if you have 5 5 data points a I here have 7 In the middle 1 8 30 at 130 megawatt that's the 1 I use yes here I have even number Of those values so I choose the the value that's midway between these 2 the median value that way but I can't use a statistical approach that
22:31
doesn't that minimizes the effect of outlets because there may be out data out there that's really not good yes but I have no way to judge in order to minimize the impact of outliers I use me you know about them and if you use this the values you find data that's actually very often use used of close to watch very often 1 can hear you have data for prostate and of course but that the strengthening elements here are except with the exception of carbon and nitrogen are different right have selected by older pytania model the the strengthened theirs and not phosphorus silicon and manganese as indicators of our fair another thing that's important it is it's here it's basically the same data you've just seen except compare them to each other so that the same y axis solid solution strengthening farright and Austinite so what do you see you see that the general level lower this season the interstitial atoms strengthening the substitution of atoms the level of strengthening in Austin that takes steals is much lower so you cannot expect In Austinite Austin fixed deals and an Austinite intended to get the same amount of strengthening that you get in farright or fatigue steel no for instance was just an example of phosphorus we have about 800 mega Pascal per precise yes there is no not a single not a single substitution all element in Austinite that has this type of strengthening effect so much lower solid solution strengthening in Austinite event in the stock and so how would how do you work you know and how do I use this this approach is very simple I have an example here so you have an idea of steel a dike Tanium so in this a composition as 30 ppm of carbon 30 ppm nitrogen 0 . 15 manganese 0 . 0 5 silicon and 120 ppm of phosphorus OK so 1st of all I think yes Is there any carbon on nitrogen in this steel yes you remember nitrogen and carbon have fuel which in fact impact in terms of strengthening In the steals the effect of carbon and nitrogen on the strength is a big 0 while I because we've added Tania the Thai Tania stabilizes the nitrogen and it also stabilizes the carbon so there is no no carbon and no nitrogen in solution so there is also no strengthening From nitrogen and carbon yes because always the careful with this so there's no solutes strangling from carbon and nitrogen steel however there may be an excess Titaniums usually In order to make sure all the carbon is precipitated this carbon coupled tightening Carbide or as a whole the nitrogen is precipitated as stating in nitrite we add enough Titaniums said we have some excess Tania as this excess tightening isn't solution and you need to take care of that amount in solution not the total amount the amount in solution the amount that doesn't for the nitrites of the carbide has to typically but this excess is about 200 to 300 ppm and and In this case will decide its 200 ppm OK so that the you would pay calculate the yield strength for instance as the the yield strength is diffraction stressed that's what what's the frictions residents are apparel stressed yes Powell stressed temperature dependent nurses thermal pod it's basically the critical result shear stress that no times too I like to use 39 but you could use 40 minutes this this experimental so but that's about the value and then you multiply the strengthening effect so for manganese I have I had in my table here political From
27:55
right rising here 43 . 9 credits this median
28:02
value OK to severe
28:05
43 I multiply with the the weight Percent of manganese I do the same for silicon phosphorus tightening Newman aluminum and in my composition by some this all up 63 can 6 3 is very long no successfully is a very low value of 40 yields rise but that's the value that's the value that takes into account the lattice friction says and the the solid solution strength it's slow but usually when you do measurement on the nite of steel there's and then there is very important to you when it's not temper rolled yes it's very low it's it's around 100 250 of that's still far away from the 63 Why is that higher in practice well 1st of all we didn't take into account any of the other strengthening effects which will discuss what I that we form precipitates when we make 13 and Carbide 15 item that has a strengthening effect and of course we have grain boundaries which act as of a strengthening also so the grain size is not taken into account precipitates are not taken into account and these will be will seep out and of course there are a few there is that there are some dislocations it's not a very sizable density would so the dislocations grain boundaries and precipitates will contribute To this yield strength also because of the debt we haven't discussed it but this is the contribution of the eye but this friction and a solid solution strengthening in the winter 63 made Nebraska OK so Of course this is a very very likely alloyed materials if I had a half a per cent of silicon and 1 . per cent of manganese you'd you'd have a much higher found get so but now let's go back to the theory that OK said so that we're going to try to understand where this is strengthening come from the and and and look at theoretical models of what 1st of all you have to know and it's a will that will focus on only 2 contributions the size effect and the modulus effect that so and as I've told you the size effect is the most important 1 1st of all what is it wants with decisive factor so it's really the pets all whether we talking about an interstitial atoms or substitution and it also depends on whether we talking about alSahhaf Byron organic would we talking about for Riddick steel or Austin at extreme care so let's start with interstitial atoms the cadre in so let's look at an elementary volume yes of the size of a Of the lattice the unit itself in the lattice little elementary volume yes yes and we assume now that we insert yes a carbon atom in this position that's in the lattice then we find out that this spherical volume is distorted Intel an ellipsoid yes so it becomes stretched in this direction and it's compressed into 2 perpendicular directions this kind of lattice training is called the 2 tribunal distortion and it gives rise to strain ellipsoid that's what happens when you introduce carbon in the fair lattice and you've heard this before since this is what happens in reality this is the CU unit cell of Alpha when I put a carbon in the optic drove position the latter's is stretched in this direction and when these 2 atoms of pushed away these 4 atoms movement so what used to be a this spherical those spaces yes In the end the slightest becomes an ellipsoid if I have the substitution of therefore this is for interstitial impurity this is for a substitution of let me backtrack that is let's keep their I'm still discussing interstitials here but I'm discussing interstitials and gamma when you put there a carbon in the
33:45
lattice of gamma Irish yes when you consider a little space Circle lost a spherical space around the position where you do it is what happens is the largest expanse isotropic Lee yes so I don't get it ellipsoid I get that isotropic distortion and so what happens well that's the case of carbon in gamma when it would carbon now in it up to he drove positions in government I push up these 2 actors and I also pushed away these 4 atoms yes In the plane perpendicular 2 of the this action so In the case of substitution all atoms always whether it's Alpha are for gamma IRA always get isotropic distortion so either this elementary volume expands or contracts depending on the relative size of the assault so for instance here I've changed the central Byron atoms in my BCC unitcell by there no actor that's larger than it did the lattice will expand in 3 directions and I put to bed the other endemic irony the solitude atom here a larger solutes than Byron madam in the lattice substitution opposition and the largest expanse isotropic Lee yeah so that would mean that in Alpha irony when I have spent interstitial I get the tribunal distortion endit when I get substitution all I get isotropic distortion yes In the case of gunmen are when I putting Carbon I get isotropic distortion and I put in substitution of allowing I get also isotropic distortion is that true not really you can get also To tribunal distortions in gamma IRA and in Austin that fixed deals and the reason is because carbon will often forms a complex it will associated itself with another .period defect and other solutes or a vacancy for instance yes and that will give you at the tribunal distortion around the carbon atoms so Alpha are the carbon atom will always give rise to to tribunal distortion in the case of gamma IRA yes I will get a distortion if then carbon atom is associated with the substitution of the atom or would other lattice defects yes so that of course you can ask me the question will have you know it's really the which we talk about the atomic at eye level and the amount of distortions are removed tiny as to how you measure was very simple just measure lattice parameters so for instance if I take up the but gonna Austinite NI I adds carbon yes at carbon to would find I find that the lattice the unit of bill of just increases where becomes larger so it's isotropic if I do the same weight carbon but they should be but this is this is big the here please change this is an area should be for of these but what you love what what you get is and you can see this effect in Martin sites why do we need to make Martin site because the solubility of carbon is 0 in farright so you have to go McMartin site which you Martin side is that as you add carbon you you get at a tribunal distortion Of course and that's because carbon gives it tribunal distortion Of the opted neutral space and you can actually use these 2 parameters to calculate very precisely what the strains are it also for substitution all atoms it's very simple if you add aluminum to our news if you add silicon to IRA grown to BCC the lattice expands or contracts a homogeneously and so you can use but the slope of this line to compute the change of the lattice parameter with the concentration so you can
39:26
MySpace the slope of here in gives you for instance .period altering meters per the 4 at 2 case and on and for you can do the same and as you see here in this craft the you can have homogeneous isotropic expansion but you could also have isotopic compression that depends on the relative size of the the use of the actors and and so here we have a list is what is known about this but even here if you collect all the data that you can find in the literature that there is a range of values available and you kind of have to choose amongst each value notes for this delta so what is Delta is basically the the the linear restraints just off the lattice parameter with the addition of DDI lighting Allen so that is an
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important parameter and so now we need to In order to understand the solid solution strength think we need to I understand how you know what's the interaction and how how do we describe this interaction and hold up and what I did the important parameters in this interaction to 1st of all so foot for the sake of simplicity and to make the geometry simple what I have here are the role of atoms yes and and here it is the scraps gusts here market dislocations a dislocation Excuse me but has run into these these patterns with just think of them now as obstacles to us and we'll see how you know how it will have the interaction actually works and just think about them as obstacles says so the dislocation will bat bowl at out between the 2 solutes against and his bowling out will continue on tell you we reach a critical point and the dislocation is released by the obstacle yes so let's say so what what what can we measure wouldn't measure here yes so we have just so we have mind the dislocation is running through the press yes it arrives here it bows out there and it continues to bow out this I'm busy and then at a certain value all of my so community external stress is how tends to be on the dislocation has if I have a certain value of distress I will have a certain value of the radius myself as I increased distrust it will and it will book the bowl out more and at a certain critical value it will it'll be released and that there you and so that certain acid at a certain critical value yes and so what do we have would under the forces that are in place when the when we have this interaction OK so you have to think about the obstacle care and Indian you have to think about the the dislocation of so what I'm going to do it is I'm going to use it the pair scissors yes lyricism and cut this off here this amendment cut this also hear same pair of scissors and cut this off so I'm in order to keep the the balance here yes I need to have restored the forests this along with the peace that have cut off yes end I need to consider the force all of the obstacles that balances this year that what is this force will the sources nothing else than tension that's that works on the line the dislocation wine tension that's it's a constant value yesterday and which working in this direction is the forest Of the obstacle on the dislocation which recalled after note what happens when you increase their or rather decrease the radius of curvature here when when you Our are insane making the costs larger units the cost is larger this the force actually becomes larger litany draw nearer a vector diagram it's it's better so this is for have the year the year the force Of the obstacle this it has to balance this yes when the the young the stress on the dislocation increases and I have a lot more pronounced because here the is now In this dislocation line tension is oriented this way this is now the force that the dislocation has to but then the obstacle has to exert on this occasion and when this f value reaches all went with the summer reaches a maximum of yeah then the obstacle releases the dislocation that so what is important here is this bungled 5 no end if this is Max here said the path is not affect this angle here now the dislocation will break away from the obstacle this angle is the critical yes 5 and of course if the if the critical goal Is this 1 if this would be the critical yes Max so in other words if Max is this that's so this would be a week obstacles where is this is a strong obstacles that so weak obstacles have fights these are small but critical fumbles the small Strong obstacles have 5 very large yes so if I would look into a material With this location I could make a snapshot of dislocations moving through a material woods to wrong solubility this location interactions between the dislocations would look like that yeah this would be strong if I would look into a material where I have a solutes which gives me a week interactions with week obstacles this locations would look like this yes because the break off the breakaway on uncles I would be very large here the breakaway the the break uncles he had this article here would be Close to pie over 90 degrees very small angle in this case no 5 critical is all is close to 0 yes so the shape of the dislocation tells me something about the strength of the interaction of that so let's put this in a little bit of formalist here will 1st of all we have and 10 years the
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1st of all we have the forests on the dislocations so that is how times speed times the length of the dislocations of the force on the dislocation OK and he an event so and if I can concentrate on the situation here I have basically this time the blind tension going here the line tension going there some of these what and if we know that this Congolese 5 is to achieve sinus 5 so sinus fire balances the external forces against and so I can rewrite this equation very simply by using what we know team to changed the square divided by 2 and so In this this equilibrium Of these 3 factors can be expressed like this but I also know White want to force is on the dislocation because that's I applied externally right so from the Farm in terms of the externally applied force I'd have to go 20 times out and which now if we look at the breakaway moments so we have the critical shares stressed the terms about his f max and if Max's this simply this equation here where 5 is the critical break off so now I'm I write this as to how it's the cost to the divided by l times "quotation mark of the critical it accepts this is what is this this is the sheriff this year's stress applied to do this visit so that's nothing else that's coming from the externally applied stressed right so this tells me something about the strength of the material G and B art materials parameters the modulus and respected so that's a given so what added 2 important parameters L and this article the critical breakaway dunk so tells me something about the strength of the interaction of like said their fight is large it's a weak interactions fight is small it's a strong interest this that's a large weak interaction small stronger and any other thing we need to know is the obstacles facing basically if I have information on these 2 things I can't help to late strengthening at no moment In this discussion did I ever say what was that would it ever was there ever anything specific about the obstacles actually the obstacle in this particular hot chapter is our solutes but they could be particles there could be dislocations I could be anything really so this is actually this equation here yes yes it's actually very fundamental equation yes In but when it comes to strengthening mechanisms in steel and actually in any Crystal wine solid noon and it it OK so let's say
53:56
just images so you have a feeling of 4 numbers here right now so let's let's calculate 1 of these angles saying say we know but some parameter values which is the what's a typical so this this angle is determined by the is the maximum resisting force of the solitude after and Idaline tension of this of courage right that's basically so then we are going to try to say something about a solid solution strengthening so let's saying we have the uh silicone or manganese and farright now and I and we're only going to weaken to look at the misfits the effect on the strengthening so we need to use the parameter Delta yes and fought for silicon and manganese it's if you go to the last point to 2 . 0 3 it's a typical value for Delta Force and let's let's take point or 3 as the value so what is the force now for a single Solyom was single manganese atom will exert on the dislocation this you have to accept from the 150 GB squared times dealt with I haven't shown proven which which showed later but we to have some form of the 4 F max if we want to calculate the critical on it so good let's assume and actually this is a very useful formula if you will efforts In doing this kind of calculations this is a good formula 2 To get the an idea Over very reasonable idea of the forest that solutes will have on the the this location anyway so so you have this formula engine you plug in all these data values of this society the value of chain this is the value of the and this is doctor is going to calculate this and you will find a very very small forests of about 3 times 10 to the minus 11 Newton's has but it's now 2nd thing we need to calculate is the line tension and because the the feisty is the art of max divided by 2 so the line tension is GB square over because that's easy this is 1 over to this is Jean yes and this is again be square the it's so now is 2 . 4 6 and pantomime 11 new entrants so I can I can calculate fight scene this calculator here is 1 . 5 6 radiance or 8 the 9 . 66 the 18 and there's only 1 degree off 90 degrees or half a degree of nite so that means that you dislocations will be interacting with the With a manganese ore silicon atoms yes they will bend out a little bit and then all quickly the released by the obstacle test so it's it's what we call it the week interaction and that's it you will usually find us very close to 90 degrees so the dislocation will have a relatively Straits shaped in the depressed "quotation mark Anderson and so we have we obstacles week obstacles in the sense you know that you don't have these very large cups cut for him don't forget this is only 1 single actor right Of course but the strengthening comes from adding more atoms the 2 them To get more solid solutions strengthening but it's ready so now How
58:33
do we calculate that staff right obviously you can feel then when we calculated as we need to have the history the feisty the critical uncle we need to have passed into 2 years the line attention so that's no problem I only need B 2 calculators to calculate after Max that's a little bit more of a challenge here so I need to say something about the interaction between the this location and the solid so it's say for instance here is the ladders and here's dislocation and in this lattice I've put in yes but after a misfit thing after has so so that there will be around this afternoon there will be a displacement in this is misfit fitting Adam it's larger than the and so on and around it the lattice will be squeezed so I can calculate what the distribution is of the displacements of the atoms yes I can act calculated shears trains at the at these interfaces you can think of shells around surrounding this atoms and the shear stressed that applies but I'm not going to go too much in the Mafia because those matters rather the lengthy and that more interested in in giving you the results so you can work with titles but anyway so that that's the idea so the interaction here will depend very much on the actually where you put the the dislocation also but because of the dislocation goes passes on this plane yes I will get a very different interaction than if the dislocation passes a little bit away from the item and if it's very far away it will not feel the point this all you can do so at so that we so you would we will what will 1st try to do Is will try to determine the interaction energy what's the interaction energies between a dislocation and a mess fitting obstacle and came right so so this is just schematic here this is a song that here and say the interaction energy looks like this it's said it somewhere the potential well so basically this means that the dislocation is attractive is attractive because it doesn't have to be this way it can be repulsive also to the dislocation comes here yes is attracted to this potential well yes and then when you want to move it out of the potential you have to increased the stress on the dislocation to pull it out of this attraction yes so but that's not the force of course because that's just the interaction energy to the forest is derivatives all of this energy so that Of course the dislocation so did obstacle has an influence on the dislocation and of course there is a and vise versa the dislocation has an effect on the obstacle right so if I do if I want to date forests on the that the dislocation yes for some of this that I have to do minus the derivatives after the energy so that looks like this and of those standards so if it's attractive interaction the dislocation will get when it comes close to the defect it's going to be pulled into pulled 2 words but the fact that and to the point and then went when the dislocations needs to go beyond the um the solutes it'll have to be that pulled loose of the uh the point defect solve them because of that's that's the shape of the force force distance and relation here OK Rice said
1:03:08
so let's do a little bit of a math here because we do need to do this so we have distortion around a substitution of following element and it is this distortion that interacts with the stress field of dislocation and it's for this and this interaction leads to an increase or a decrease of the elastic strain energy of the lattice but so in this discussion on this series we assume that this the atoms are elastic spheres with a certain radio it's inserted into a spherical lattice hole created by removing an Irish with the Rangers already and it's the relative size difference between substitution all atom and Irish yes that is of importance so endit so we use this parameter Delta and Delta is is nothing else in this this racially the difference between the radius of this solutes and Byron atoms divided by are so it's that the radial strain if you want to you can rewrite this and the IRS is 1 plus delta are our OK and this is the important parameters this this gives us an idea of just the amount of strain that you have a volume strains but it's a linear parameter has so you can calculate a volume difference yes between the solitude and the solvent at and because that's the volumes strain that's important here has and you get what's called this misfit fall you know that's basically the basic calculating the the change in volume yes that that needs to be accommodated in the lattice so that is given here yes we assume that all actually we know that Delta is a very small number so we can we can simply do this simplification here and that gives me this value for the misfit volume Duncan that's so if I have an isotropic for defect that I get this Miss states a large it's large right you will have a hydrostatic pressure there was no indication hydrostatic pressure yes at the interface between the substitution of atoms and the iron matrix and so I get elastic strain energy elastic strain energy now then which is which is equal to work by pressure against the volume change to this this would be energy so this Peter times this Delta the Biden said at this stage with this pressure is such as missing Peterson so that is elastic strain energy so now let's backtrack a little bit here and do we have In hydrostatic stress associated with this location from there yes we do with the hydrostatic stress is the minus In this particular case we use minus because it's it's easier for us to work with Sigma expects the Sigma why like sickness easy divided by 3 where these are the principal stresses you remember this value right that's that's the In the theory of plasticity which seen that's right that's that's the the stress along the the yields service your means stress yes so we can compute computers let's have a look at how much to assess how how you how would you computed as well you look for cinematic Six Sigma signals easy in In the previous lectures yes for an edge dislocations and these are the formalist and if you make the sum of the soaring and you divided by 3 this is what you find this is the the Hydro static stress associated with the edges and but it's a function of the the position has been in space around the dislocation so that well you why don't we put some numbers are right and I said this is the formula that was on the the previous page for reasons be clear in a moment and it's simplified fighters bite USA the instead of X and Y I'm going to use that X is equal to "quotation mark Teeter directions why is equal to or assigned Teeter has just continues this transformation it because it simplifies the before militant this year so In this is why divide by squared was white square now becomes signed teetered over our heads and and that allows me to to compute that this pvalue simply so let's say we're looking at Alpha we have an edge dislocation here and we want to know what what would be the maximum the maximum the hydrostatic pressure I can get a bit rough around the world call while about while all just choose feature equals to 1 that's a serious teacher equals to 1 that's that's definitely going to give me a large value I'm going to choose are a certain distance away from dislocation of about 10 B the substance and energy it is 81 . 5 the capacity so that I can calculate and I find 1 . 6 meter Posco that's a pretty large pretty the larger the hydrostatic stress "quotation mark so what happens now I have a piece of material that wants to expand on 1 side and I have a hydrostatic stress somewhere else yes that can provide this defamation yes but if they come together yes b energy yes can be locally diminished yes by passing these the hydrostatic pressure provide the work 2 expand the lattice so the interaction energy is the hydrostatic pressure provided by the after the dislocation times the volume Change and so is basically what we the rise just now the Times so that is the hydrostatic pressure and that is the Delta the and I can I can express this in terms of X and Y or in terms of R and B the uncle Peter so right fighting also used here instead of using it this Our square 3rd the radius of the iron atom to the 3rd I can use a Delta the parameter Delta the parameter is is given this is nothing else in the lattice volume change you to 1 solutes it's it's it's this here you can see for Delta this so it's it's an an alternative way of using this the same question let's OK so let's
1:13:01
just want to take 1 more minute of Joseph for instance would assist me I have manganese interacts with an edge dislocation of OK so this is my interaction energy I have all these parameters yes I know would years I know would be years I know what it was all is I know what the Rangers is of R my Wiranata and I know Delta for manganese so I plug this all into this energy equation yes so the energy is now here suspect this is result this is the interaction energy it's it's it's a potential wealth yes and by now computer for instance in this particular case and will compute computer force on the solutes yes that's a derivative of the interaction energy so the derivative of this line here is negative of course it reaches a maximum and amp but as a minimum and maximum and so so this is the force distance relations and in this particular case it's we've chosen why is equal to be so why is the unit and the distance between the the position of the atom is is here and and and the dislocation moves this way but at nite so soon you can see obviously that depending on where would this distances I'll all have another profile right but I can calculate from these different profile what the maximum value will be yes and that will be left Max yes anyway we overtime will continue with this on Thursday
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