Mechanical properties of steel 9: dislocations

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Mechanical properties of steel 9: dislocations
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Cooman, Bruno C. de
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The nineth in a series of lectures given by Professor Bruno de Cooman of the Graduate Institute of Ferrous Technology, POSTECH, South Korea. This particular lecture is a detailed exposition of dislocations and their role in steels.
The Graduate Institute of Ferrous Technology (GIFT)
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so Leicester right sorry fortunes during the summer of difficulties to get together but I will have a makeup classes I guess 1 of these days we'll talk about it on Tuesday weren't so right now this is a little less word on the way to the signing of much would workers factor was the house was related to this extra half playing and that indeed it is just a conventional the way defined the direction of the on the burgers factor and at this stage basically you can think of the Burgers factor as being on the amount of shift lattice shift you have to give so that you can compress the Beyond the extra have blending into the lattice and the at the end of this line as I say was the end of this section happened at what we call the dislocation lines and that wait already said that dislocations could this August 1 doesn't have to be strategic interests curve around In the the crystal and then go from a thin edge orientation to screw orientation and and when it does that I whereas in the edge orientation can clearly see an extra half plaintiff if if you would look down the lattice and that's actually what what happens but you if you if do dislocations takes a turn a 90 degree turn then but you're dislocation becomes a screw dislocation and there's no extra have claimed to be seen anymore spoken and the other thing that yes we said is that before I continue obviously you you probably already noticed that in the case of an edge dislocation the at the British workers perpendicular to the this location life and that that's the end of it the case of the screw dislocations the dissipation factor is parallel to the dislocation life OK and that you know if you're ever confronted with having to on the side you know of dislocations of where the extra half playing comes from if you have a dislocation and the school orientation you don't have to worry you just make a you know just think about events in your dislocation and the burden because the factor doesn't change and the dislocation line direction doesn't change either you or you can easily find out using your the right hand to rule on what side of the the extra half plane as an advice illustrated here it's 0 in steel you have to say you have a steel piece of steel and uh you know that contains many grains and grievously single crystals so on you can you can think of that inside the 1 grain of steel on location will will will see where dislocation loops look come from but there is a a different processes by which we can make dislocation loop but you will see that the very important 1 in the case of of steals is the Frank Reed socks and so anyway we have a little that's been created in this town Crystal and we have a Burgos factor we have the direction here so we look at the edge parts the exports are where the dislocation Burma's factor is perpendicular to the line direction so on 1 side the dislocation will be in the extra half plane of the dislocation will be coming from the top and on the other side will necessarily come from the bottom so the dislocation loops can collapse on itself there but this is occasional can become smaller and when these 2 I meet the dislocation will will have disappeared basically mill will talk more about what it all means forming mechanics in animal but so 1st of all we we won't go into the match here it's not really necessary the dislocation course there's a around the dislocation court already for many years people have known what the stress field looks like that and so do the stress field is depends on to physical parameters the the shear modulus and the Burgers vector so and everything and I remind you of the fact that what what we spent some time talking about elastic modulus is modularity and share the particularly the shear modulus but well that's that's 1 of the reasons because it in this
location theory always pops up but rather than Young's much focus of this by this primary is important so if you doing calculations of different temperature composition that truck so you've got to and then you have 80 a geometric parameters like this indicates the screw dislocations it's X divided by the square plus white square so I would say we will go into the shape of the stresses that at this stage but what's what is important is that in the case of the as good as location would have Shearer components because have mixed indices whereas in the case of the edge dislocation I also have a tensile components right and so will we know how in addition to ship off get of In other than that we just call back in a moment of what you see Is that all of these equations have in the denominator X squared plus quite some form of EC squared plus 1 square experience was that we work where and you've also would assist means next to them next year if you have a recording system and you look at a point here which has coordinates x and y right by the distance 6 square plus most white square yes the road of this so it means that when you go away from the dislocations you have the distressed feels decrease as 1 over the distance right so it's what would to basically means this distress field will be close to the dislocation and then die off it was in as 1 of but that so now dislocations are interesting defects will talk about other defects as as we are as we continue this discussion on this chapter here but dislocation are interesting in a lattice defects because they're they're non equilibrium lattice defects In addition what's wrong with would winning it will take for instance vacancy but you you know that if you have a crystal and you heated up at a certain temperature there will be an equilibrium concentration of vacancies inside your war Christopher we are and you know as you increase temperature reaches a temperature you will make you will have more vacancies at high temperature varies smaller amounts at lower temperature because of this location it's it's not like this you really need to deform the crystal to introduced them to the thermal energy is is not enough to create this location OK so I will just stop and look at the energy of a school this location because it can be very simply computed so you can think of a screw dislocation as you remember you we cut crystal and we shifted that 2 parts which respect won over by vector of the size and yes images of this is the same thing here to Bissell of looking at at the Crystal I just look at the cylinder of material where I can do the same OK so I shift I have originally it a nice cylinder by making cuts along here and then shift the bottom part bye a distance be can now on it it would look very up and and so obviously when I do this you have a the this cylinder you know that you are shared will want to jump back so it's got some elastic energy stored in their right as of the way you calculate this year's by noticing that if you unfold this if if you would unfold this the unwrapped as it were this the cylinder against you it would have been originally it would have been a flat flat volume of flat rectangular volume here and after the shearing by being yes by it's it would look like this it would look like a sheared the rectangle thank you the elastically shares and so which would have a tendency to jump back in this direction so you you can calculate the the energy that is stored in this uh elastically sheared fur rectangle so the the energy the elastic energy of the screw dislocation so it's the sheer forests times the amount of sheer and then I need to to multiply with dozens of the volume of the material in the the fall you hear that I'm sharing it so you can already see here that the amount of energy I have will depend on the length of the cylinder longer two-cylinder I have the more energy power will have 10 so so I need to multiply with the Wangs yes and DEA here so I can plug in the and the parameters one-half share forests that is the segment ex-wife who went to the White z so that would be this
parameter here but in this case but at times the
the sheer that's and be devoted to apply our is the height here of my unfolded cylinders so that its circumference this leads to buy but times l times DEA report that the ASD Is this the dark gray area the so that's by art plus Deora square miners pie are square and only keep the terms the alright terms in Dr square are small enough to be there set equal to 0 so I get an equation here that says the energy for this piece here it is stressed and the Burgos worked times the length of his DEA are divided by 2 so OK that obviously when we do this energy calculation there are there the 2 important things in things you because if people want to total energy I I will have to integrate all 4 which is basically the distance from the core of our equals 0 2 c to somewhere In space away from this dislocation right so there is here and with the computations Of the energy of dislocation a little bit of a problem and that's the choice of these boundaries you the minimum radius and a maximum range as it turns out that you can't set are equal to 0 when you calculate the full energy because this uh these stresses that we have derived mathematically they don't apply to the core of the dislocation yes that's 1 thing and then the other thing is it's where does the cut off and you know why do I say this is the end of the crest of the convicted person very large and so how can cut off this integration at very high distance came out so what people do is say Well you know if I don't have too many other defects the limit of that integration is kind of the end of the crest right that so that's a reasonable way of doing it and and the and the court will just avoid which is devoid talking about us when we do this integration will say there is that we have cut off limits and and will integrate between these cut off limit so we have an inner cut off limits which is . 2 2 1 nanometer I remind you that so the lattice parameter all of them Of course the list is around .period 3 yes but it's it's about somewhere between is somewhere slightly higher than a lattice parameter that distance weak around the dislocated that we considered the core of the dislocation where there is a heavy lattice distortion and where are elasticity laws don't apply and then for the I want cut off distance well we take the end of the crystal but most of the time we we say well in 13 steals art we have a lot of dislocations so this is locations will interact with each other said at the end of our integration will set out the distance between 2 dislocation so I'm so when we compute know we try to compute the energy of a dislocation as we it it well it will consist of elastic energy and core energy so you total energies the core energy which I'm not calculating and the elastic energy which comes
from integrating this equation here in the case of a school right you don't have to do this as people have done this and and and and and looked at it in In the past and so and this is this is what they come up with such diverse lacked analytical about the formula relatively simple again our share of total energy is caught the energy plus elastic and energy GB square over for so where should Jesus shear modulus and then a factor here where we have .period ratio yes and that an angle valve angle alpha is the uncle you have between the Wine direction of the dislocation and the Burgers vector there so so if I have for instance a very simple dislocation Wittenberg respect here yes and I defined this as my a unit factory along the direction of the line direction so In this case but the Alpha angle is 0 yes In this case the Alpha Angle is 90 degrees because so the the 2nd term here is relayed To the type of dislocation we have look then of course comes in term here L which is the length of the dislocation of the more length the more energy and then the factor of this comes from the integration of the of the year elastic energy is ln are divided by our 0 plus 6 and this factor C areas basically the core contribution and is about to you can right is about to and L and R & R 0 are basically the these integration limits I was talking about these cut of values are 0 is somewhere slightly larger that the a the lattice parameter of and are there as upper cut off is around 100 enemies you in a moment how would you can calculate and so and so this equation really holds between these 2 limits since then right because then at the deal is the lastest city In the reason is because of alleged that the elasticity theory only holds between these 2 limits so that would winning France's if I knew of a plot the I didn't to shear stress that's surrounding but a screw dislocations right 10 as a function of distance I just told you because as 1 over the distance OK so goes as tout IX if I only look at 1 plane and so it goes as 1 over X so the so so the shear stress would go like this like this Black Like right straight line in a long long long however In the course it's not us is not as high the stresses the shear stress and also at the big distance it's also not that hard here is because there is a breakdown of elasticity on this side it's because there are other defects within that adapt Rangers look let's see for instance if if you want to know what what you want to plug in
for big here while you can say it's steel but we probably have this locations all around even if it's well we crystallized sewer are value will be half the average spacing of dislocation the will will love we'll talk about this area as as we discuss the dislocation density but sigh your dislocation density it is tend to do 14th hurry the 2 of multiple tell you in a moment what terms How come you have this very strange away all of the defining and density in meters minus 2 but that's a that's your dislocation density then you can show that's what I did the distance between 2 dislocations if you know the dislocation density is the distance between 2 dislocation will be won over the square root of the dislocation density and that and that which we take as for are we take about half this test that because that would give you an idea of how far Out is the influence of a of the stress field of a dislocation reaches if I do this well such as blood in this tentative 14 but perhaps a meter minus-2 and this gives me 10 minus 7 meters divided by 2 and so that's 15 enemy so it's actually much smaller than a typical brain size but the whole grain sizes will be in steals will be between headed to maybe 20 microns and then you know you have some of the grain refined steals which made me was lost 7 to 8 microns at all you know that this is the a distance of the influence as it were of a dislocation is this much smaller than they the size of a crew grains of it's of the Order of 100 100 nanometers a good guess 1st gas if you don't dislocations were just just a moment here about dislocation densities why do we have this strange unit leaders minus 2 that is because a dislocation density is the number of dislocations of per unit volume OK so let's simply just want to simplify things let's side we have grain that's that's perfect that's a few grain has weighed the the length here of the science is 1 meter right so in total here have accused meter and and for some other reasons we've managed to Have a crystal with only edge dislocations and all these edges are very straight restraint and they just go this way they go from this side to to decide well what is my dislocation density for this particular case progressed while and I'm saying that 1 too fires 7 dislocation I have 7 dislocations in a Q all 1 of the 7 dislocations In a queue of a 1 cubic meter of the length of these dislocations is 1 meter 1 meter 1 meter for every dislocation from so it's 7 meters of dislocations In a Q has so I have here 7 times 10 to the minus 2 dislocations that's my density of dislocation have 7 meters of dislocations per cubic meter if you know this on this unit doesn't look strange animal care but but so well it's always good to have certainly went down you get a formula to love give applied in numbers and CEO in Ojai if if you ever want to play
uses formerly what would you have to play again so let's look at this formula again this is the total energy of a dislocation it it as it takes care of the the core energy and the elastic energy around to this the formerly just song and so we just go through all these parameters tried to find out where we're at and so on 1st of all well Our modulus we take the room-temperature modulus for far-right for instance that's 82 bigger Pascal With these formulas as with any formal it's always be careful with the units using because you know of that needs to be changed usually the Giga Pascal things you always have to change and a new terms the meters the being in these burgers spectral show you how we calculate this burger Specter for the BCC iron but at this stage is take it from me it's . 25 nanometers and we did so again I turned us into meters that's a point 25 sentenced to be this should be a minus 9 by the way if you up if you made the copies already make sure this is minus just notices and we take that we look at that screw dislocations side so we take Alpha equal to 0 and we take L equal to while you we have to make a decision right how long are we going to take this this look at this as we will let side were looking at a tiny bit of dislocation very short bits of dislocation actually it's as it's as long as the Burgers vector them just remember that it's a small amount of this small piece of dislocation said the outer cut off a radius we had calculated on the basis of our a dislocation density to be 1500 are in the cut off we go we said something that's slightly bigger than the the unit and then the unit-cell dimensions of but the the way that or a value that people very often use is 1 . 5 times the burgers factor can't see as I told you and take it from me without the explanations is about to this parameters and of course the because we like to express things in the electron volt analysis of all tell you the moment why and also give you do the relation between 1 electron jewels so you just plug all these numbers in this formula right and then you find a 3 . 5 6 Evie there's progress but so what is what does this mean that on that is that you know is the future of the shouldn't you will be impressed is as large as the small for us what's important is to relate this to thermal energy yes so I at room temperature indeed the because of a lattice vibrations because of temperature we can supply there's a certain amount of energy 2 defects in the crust that and that's the thermal energy and have visceral energy is so low that we know I can be calculated it's just basically bolstering constant times the temperature OK and I said do it if you if you calculate this fall room temperature for 20 degrees can you find a value of 25 million electron volts good value to remember yes I'm so so it's full . 0 and I'll to make sure we have electrons can I'm right so you can see that this is a man of many times smaller much smaller than 3 . 5 sex you think so you cannot create With just thermal energy you cannot generate a dislocation even very tiny 1 against and because this this value will will cost blowout as as you increase the length of his dislocation and here I only took a very short segments of dislocation to dislocations in contrast to for instance vacancies are not created spontaneously in In in crystals cups but so we
basically have to perform material if we another thing that's interesting is this to look at the end ST the energy of this focus is at constant so for instance if if I look at the energy of dislocation on this part of the dislocation and on this part of the dislocation instead of constant and this is what will it turns out it's not and you can see this because there is any but the factor that's related to the type of dislocation so if the so if you look at this parameter for a screw dislocations Alpha is 0 so that means the co-signer of Alpha is this is 1 and so this parameter this factor here is 1 however for a edge dislocation the Alpha is 90 degrees so this 1 over 1 one-liners the racial OK so you will have the difference in the energy Of the edge dislocation and the school dislocation and the use of you go the calculations you find is that an edge dislocation is 1 . in energy of images have been 1 . 4 times the energy of his school political so that means that it because they're dislocations In general the dislocation loops have higher energies for edge this location relative to squeeze this location notes will always tend to be elongated With longer screws segments then edge segments In order to do minimized the reduce the energy of dislocations did so some some some general thing is today are important to 1 of the things I want you to notice is that of this factor here the natural logarithm are over our 0 plus see it looks like a factor that may have a big impact on you know what the energy comes out actually doesn't the value Of this factor you know for reasonable races of our overseer is around 5 despite fish has to then it doesn't change by orders of magnitude so that's an important thing that we the other factor here with the co-signed is related to the directions of the dislocation and so many other stuff is basically Constance G impact so but we see an important thing here is the beast square the energy depends on the will the magnitude of the bird respect and so on and will see this Back then in a moment when we consider also the y intention of dislocations conceded who introduced them the yeah parameters called line tension which is actually directly related to the the energy of dislocation and that's the line tension and an underlying tension is equal to the good approximation be squared divided by 2 but so what 1st of all let's talk about the fact that the energy of the city depends on them the value it and it basically means that our the this locations will always get configurations With b square values and to reduce the size of the energy of the look and so on what it would accept it as a consequence well in Allston epics stainless steels Nora Austin epic steals all that you remember desires deals with an FCC structure and here I have to say with the lowest stacking fault energy the fact that the energy of the dislocation is proportional to the square this means that the dislocations this social so that means that a dislocation for instance this edge dislocations here this is the the line direction this Burgos factor and just forget about this the symbols for a moment here what you will tend to have is the dislocation will dissociate into what we call to partial dislocation the partial dislocation With a stacking faults between them and the reason is that 1st of all because there are but the variety of this dislocations Burgers vectors are possible in the structure of and they have different sizes but what is important here is that this dissociation happens because the beast square of this Burgers vector is smoke is larger than the beast square of Deese too yes but you get this dislocation dissociation but the fact of course when you get its association is the creation of a 2nd fault and that and of course that is extra energy because when we create a stacking fault we we stop there normal Crystal Graphics backing and we increase the energy of a crystal of the dislocation this way so there is a balance to be had here but will talk about his work In a moment of but now so just for your information and will talk about this at a later stage which would I'm using here to describe the dislocations in the FCC crystal Ike Austin seals is this so-called Thompson Tetra he drowned and and will give him a demo of how to use this but very short because it's it's not really central to our cost but all show you how how it works OK so what but would would basically have is that this is edge dislocations sea the yes it can be dissociated in 2 partial dislocation see Delta and they'll be and and you could see the smaller in size then CB and have its property that the square of CB is small is larger Excuse me then
beat them the Burgos structure for Delta and the Burgers vector for Delta sued both squared and stuff like that again will will come back to that but it's important that isn't in relation to 2 introduced is already at this time so you you aware of fact this locations can dissociate and and they do this because of this we square dependence L Red Cell and another thing that is of interest now it is say we have at this location yes In aid In in grains in a steel yes and look at that and we knew we which said that defamation plastic deformation permanent deformation is the result of dislocations moving through the crust and and how do they do this how do they do this while it's because when you apply an external force this forests In some ways works on the dislocation makes the dislocations moved I'm just going to give you a simple example here right so sigh this is a Krystalle years this is a glider planes the dislocation to the crystal the information is by slept now so if this slipped is by single dislocation knows for instance like this so I watch this crystal inside itself a push it inside itself so that there is an extra half now here so when I pull here this dislocation moves to the this and comes out of the crystal and I've ever have a permanent deformation well so I this applied force works on this dislocation it so how do you approach this mathematically so if you want to know you know what's the effect of it applied force on the dislocation well I'm that's been worked out a while ago and on I just want to show you how it works so again this focus more of an influence of externally applied force them into what we need to uh consider if we want to calculate things now 1st of all we need to know Barbara of dislocation comes as no surprise we also need to know what is art unit vector along the dislocation lines and and we need to know With distressed status 1 like in the case suggests drool it's very you know it's a simple stressed they assume the actual tensile stress right but I'm in general and you our stressed it will will have more than 1 component To tensile components and sheer Compal morning but it's so let's let us I have a look for instance let's not apply any force on our crystal which is let's find a difficult situation with which which can easily be calculated for instance if I had a screw dislocations in my crystal yes it has a stress field around the it has said so but this matrix as describing the stress stayed in the case of a screw dislocation is this 1 because I only have shared components the X Sigma X X signal y y Sigma is easy are 0 in the case of a screw dislocations and and the values of segment ex-wife the incident the White Sea are also say I look at the screwed up at an edge dislocation and this matrix here it is like that OK I have a Sigma except signal y Sigmund easy and Sigma ex-wife I always we're in this house just wouldn't just want to to say this in this case it would simply be just 1 comport segment expects practice so but I prefer to give you a more complex case because it's also more informative than now the the force that works on a dislocation has 3 components 1 and x y and z direction where we choose the x y and z direction in this way here on the Net make sure I have this right so the agency direction is along the the dislocation lining the 1 I is parallel to the slip plane normal yes and the x axis is perpendicular to the that makes sense to choose and talking all basis on on the market right now the forests and is there is calculated by doing this is calculated by getting there 3 components 3 components of this affected the X component complements the component and these other former officers friends FX is Hermida a wide times to easy miners ACT times to watch T D the coordinates disease coordinator and the white coordinator of the 1 lying direction in this adjustment of the line direction and Maine X the wide and he's zooming are given by these formulas here who's used to friends a X is segment X X and B X you know what makes white extended BY Sigma z x might be times piece and the same for 4 that's something similar for the wide and using this is this is this whole sad to hear of equation that's the actual peach crueler equation and then it allows you to calculate for some this locations for any complex or not complex but the situation
you know probably this short form of the peach-colored equation years it's usually the simplified and compact form of this equation that applies when the dislocation geometry is very simple so these it's important that the to remember that because the Burgos factor is always the same for your dislocation the force under this location will be the same everywhere but its direction may be different for instance if we win you when we did forum at this look bad actually a single crystal or grain in the steel samples what actually happens inside the grains on loops of dislocation expanding this expanding so there will be parts of the dislocations which an extra have blamed pointing out for some other parts will have this pointing downward and the dislocation this location here will move to the left and this 1 will not host of a move to the right otherwise I won't get plastic deformation right so in other words the has to expect right so does the forests on these 2 parts is the same but not the size not the direction that allowed makes it possible for to expense because it is important and also please remember that this is a the young and it was simplified form of the peach-colored equation this is actually what you need to do that at no cost it's all nice and easy for me to say but it would be nice if you had some examples because of how you use this speech Corolla equation in practice that will say for instance you have to say screw dislocations that 1 screw dislocation here and another screw dislocations somewhere at a distance In anyway and all this 1 no 1 has a stress field around the yes and I'm trying to figure out what is the influence of this stress field on this dislocation in other words what is the forests working on this location number 2 Due to the presence of dislocation number 1 so what I do in this case so I have the screw dislocations going through but going along the c axis remember we in this peach color formula we we take the z axis along the dislocation wide OK and then some this symbol here it looks like an anticipation that I use it for just symbol for this location for a while but it here in this case it's an assistant that screw dislocations and I know it's as good as because I've taken the the unit vector along the dislocation line parallel to the border respect the purpose factor in this case isn't necessarily because it's so long as the axis 0 0 BC in again and then I have on other dislocation yes at a distance away From of the of this 1st school dislocated it's also screw dislocations and science said it's at the height here deed from above in the y direction but this is the blight plane the best and this is the best and the brightest it's away from the original it's not unnecessarily on the on the same glide right so the so the Burgos factors of these 2 dislocations can they can be the same or they can be reversed right so if the burger Specter artist saying I say was the 1st Burgers vector 0 0 B 1 0 0 B 2 4 the 2nd 1 and 0 0 disease is the common burgers for what they have can have opposite Burgers vector this 1 this case be 1 it's 0 0 plus to easy for instance and b to its heroes reminds me of that right in the unit factor along the both dislocation lines just this 0 0 1 that has you simply by and distressed state of art dislocation here is very simply this but this 1 here this equation this area matrix where I I put in the values for the equations rather for Sigma X Y and Sigma y z so you get this nice matrix at and no and now I churned by just goes by just simply go through the procedure very simple the 1st we determine these 8 parameters but this is the equations that I showed you and I plug in the values for the signal x x and Sigma X Y and the values for the the BX BY busy so and this is what I get the X is this the Why is this and easy is because of the
next step is now to determine the Suite the coordinates of the forests effects wide as the debts again by you just apply the formula that's a wide times cheesy minus easy times the Y uh OK and I put it in the values I had 4 TEXT wide TC and cheesy 1 in this case 3 why is 0 and T X is 0 so I end up finding very simple equations for FX FYI and its yes these are the 3 components of the as for suspected yes the that from 1 screw dislocations on the other 1 where you are the 1 we have said is at a distance is the from the still OK so this forest has this sex component this why component and knows he composed of 10 or so then when this but it is positive years when this is so when you have the same side they both positive or negative yes then there were they will always be pal each other the ash value will be positive because it will repel each other if if the components are negative there is an attraction and so that the screw dislocations always attract each other if they have different Burgers vectors but they always we pal each other if they have the same Burgers vector always we propulsive when you have the same size always attractive when they had the opposite side let's go to the case of the Jews do the same thing but now we imagine that the 2 dislocations are edge dislocations we have an edge dislocations here an edge dislocation that the only thing that changes that the only thing I think is that this is an edge dislocations so I haven't other stress field has stressed field of energy and instead of having severity of the please correct this here but this has been the B factors are Of the had been busy components is 0 that's B X 2 0 0 the ax 0 0 and here -minus B X 0 I made an error in obviously 2 I I will correct this online here but that's obviously the acts for 1 and for the other 1 that's correct in the text here but anyway that's the only thing that changes is that only change that to think stated to me that this this stress around the distressed around the dislocation and the bird respect it's so let's just look at that hour applying the peach-colored formally same Burgos factor so here it's correct and B 1 0 0 and B 2 0 0 is a B x 0 0 or opposite Burt perspective we want this blessed be extended the 2 is a minus Pierre the but otherwise we so we will look at this situation or at this situation yes but otherwise there you know the position of this other dislocation is so pretty much any X Y value is OK but the unit factor that's the same it's school 1 so we have this location line along the z axis and distressed state of this location 1 is just as basically in this the complex In this onto the matrix we just replace x y y x y z then x y y antics that seemed to signal x x Sigma X Y signal y y signals the with with the formalist and these 2 constants I put before the which same procedure you determined 1st E 8 8 parameters yes give you this and 0 4 KY it's scene and then you determine the 3 coordinates of the force factor With the former just straightforward application of and you find it and the rest of the component has a force in the X direction in the wine direction an in disease direction so
no you can look at
this the this interaction
when they have the same size and different side and so on so and what the usually put represented years in the following way you haven't and y axis where you I think the forests yes 4 starts this year so that's the magnitude of the force is this if you have an aspects and have the wide and effects thank you the magnitude of the force yes it is so you can plot it is a
function of X over 10 so the distance in In the glider plane and why this distance from the glide place we can if they have the same size the interaction goes like so that means the 2 dislocations look like this you have the forests yes it is we repulsive yes and then when the next parameter gets smaller order wine Porter gets larger it drops and it becomes 0 this in this situation X is equal to y right so if X is equal to wine this look the dislocations are located at 45 degrees to each other because so then I have no force so that is the the point of stability rights of the dislocation it is in this point there there is no way no force working however if like I push it a little bit further yes a little bit further closer to each other because of soaring oil that goes on in this direction since then the interaction becomes attractive again continues to be attractive In fact yes I it can be very low become very long so in this case I have a configuration that looks kind of like that's the 2 dislocations With the same prospective on top of each other if 2 dislocations have on opposite sides like in this case we get to reverse force distance the situation there but this will take us a little of continuing with this will take us a too far for the time where we're now and so on because I'm already over time will just keep this for next Tuesday thank you very much for your patience and I'll see you next Tuesday
and I'll I'll make the correction to this drawing invite today
perhaps wait until a few planning to printout the content of the of the file on the on the class wait till I and this evening to do too
this but it is


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AV-Portal 3.13.1 (abea844c86ad1b15ca76e1472346f3fd8bea123a)