Mechanical properties of steel 2: Elastic deformation
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Title 
Mechanical properties of steel 2: Elastic deformation

Title of Series  
Part Number 
2

Number of Parts 
24

Author 

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. 
Identifiers 

Publisher 
University of Cambridge

Release Date 
2013

Language 
English

Content Metadata
Subject Area  
Abstract 
The second in a series of lectures given by Professor Bruno de Cooman of the Graduate Institute of Ferrous Technology, POSTECH, South Korea. This particular lecture introduces the general concepts elastic deformation.

Keywords 
The Graduate Institute of Ferrous Technology (GIFT)

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29:49
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some reason the got title of the course and member of the course wrong all nite on Tuesday to the courses 6 6 9 and its mechanical properties of steel than other terrorist allies politically and Joe there were some minor issues where people told me they couldn't register for the class has stepped over the long has that been solved because they apparently only 10 people were able to register as everybody been able to register and then Tuesday's introduction and today's lecture it should be on the clock "quotation mark if you if you want to to have the material is available but so today will be the new when it when we talk about fundamentals but we're basically going to want to its particularly to look at what those are mechanics tell us about the elasticity investors state we want going through all of the theory because if you want to go who the theory take courses on mechanics of which will do this and see how it applies to 2 steals in real life and and and and see and how how how we apply it in real life just a reminder to start with elasticity what time what you basically dealing once in arm in steel on the atomic level and you have a solid there is defined that hangs together the rule the metallic ball right Kelly bond is something very specific and that's the difference from the ionic form covalent bonds notes an in In steel and in particular in in in the end variety in the way you you should think about it is that you basically have violence Byron yes and then you have electrons that the cost of the metallic ball that we were going to do it the right to quantum mechanics of the all but the irony is a transition metal cans and the way it works is you have basically ions of Byron which missed a couple of electrons because they shared use electrons with all the other at and in these electrons are actually for as free electrons of form kind of gas was it not localized the localized and then In the inquiry city Ferrari you have closer by still forming a a cloud or the 3 d electrons than all the other electrons are basically on and on their original atomic levels as if nothing had been going on with them when the ball was created going from Athens to the south and you have a solid core went this and if so so the if you want if you preferred this picture here which you have the floor right to have a new unit cell here the unit Little killed here if you count the number of atoms in this units to iron atoms in the solid you basically have to I don't spiral ions and you have these the localized for the best level free electrons here and you have about 4 of these electrons per group you know so basically meaning that every Irish patent contributes to electrons to the the electron gas you can you know just from this simple picture you knew there are about 10 to the 17 times and the 20 2nd electrons for a few centimeters of free electron spin yourself with that so 2 what was so what happens now is that when you have the metallic boundless created you I have the specific distance between the active this time and you have at this distance is sacked by these interactions between electrons the interactions between the violence and the interactions between the ions and electrons and obviously for instance if I if I'm trying to push these 2 the ions toward the charter no but with them too close to each other and a there's very very large positive charges will push them apart at so of the comes as no surprise
05:51
that when we look at the energy you are you're crystal you find a minimum this a good start this is the energy at 8 specific density of interatomic distance this is a I calculation which is gone the 1st principle called the FT calculation for a wealth of farright at 0 OK and for Austinite its European which you see is that there is a minimum here minimum where the wood which will basically defined what the interatomic distances in your crystal and at that minimum you but you also have a certain energy and you see that the energy of farright is lower than that of the Austinite the FCC lattice and as a consequence of what we see at room temperature also is the stability of rather than most of there is a profound reason for this which is very complex management which is related to the magnetic properties of fairer and Austin Austin and does not have an apparent visible magnetic properties but they exist and that causes the right to be the more stable phase at room temperature of and to lets there get on with things so you know that's there is a certain the equilibrium distance between the actors and if I bring them closer together they will be repulsed by each other effectively taken too far apart they they will have a tendency to come back together so in other words you can think of all this bonding stuff as if the atoms were connected with some kind of spring yes ends the something interesting this function with describes this and his behavior of the the energy of the process as a whole the atoms support of wishing that telescope and interatomic potentials and has this is kind of this kind of shape here has had people have been calculating this for irony for many other solids yes because for instance if you want to do very big calculations of the atomic processes like Clinton's in the moleculardynamics you need to have this function you need to know what happens when you move an atom closer to another 1 in what will this afternoon to them will will change its position on 1 of the very simple things you can illustrate with this interatomic potential this is what for instance thermal expansion white white solids expands when you increase the temperature for instance but it it made me do this because this interatomic potential is as matter so for instance if we so selected 0 OK if you look at this as a potential energy curve here the items will be sitting at this this specific distance here a specific distance here which is the minimum as you increase the temperature the the crystal starts to vibrate the atoms these days the springs are being pushed together strain which together as a result of what we call the quantized lattice vibrations which would call for answer and the amplitude of this vibration increases so the the atom vibration goes from here to here and from here to here as you increase the temperature from year to year as you increase a what you see happening because this well this Aceh metric you see that eh the mean value as ornamental value as it were of the atomic position increases so the thermal expansion is due to the shape of this interatomic potential doesn't always have to be a solids which was contracted with when you increase the temperature or that they're not expands when you increase the temperature detected a very famous a Byron alloys it called in Nevada which has this property and the reason why it that you know you have negative or no expense thermal expansion is again due to special magnetic properties in the case of steel there are involved in it and and they're all undergrad expand solids which which actually can can contract with temperature but that's not of our concern today so In well let's look at 1 of these interatomic potentials because of these functions this is an actual 1 here so that you have actual use these here and you you see here this is your minimum then there the function increases and potential energy becomes goes to 0 as it is in a very long distances but In this particular irony Irony potential that I've taken from the literature should be of any concern to you at this stage but the minimum value for the interatomic .period 26 2 now I don't know and that's very close To be for instance the Byron Byron distance but of the points the 224 8 nanometers in the 1 1 1 direction between the 2 firing at that stage thank In that lattice that's the the distance of closest approach but so if we have the potential the energy function like this so this is a potential energy prejudices in EVE and if I take the derivative of dysfunction so 2 effects this is EVE and this user nanometers nest and Evie is energy it's like work that's the work is you area forest times this the new Tom's Fernando meters yes and this is an enemy so when I take the derivative of a potential function I find Forbes by the end of the the brevity of this interatomic potential allows me to determine right away what is I will experience when I push the 2 items together or pull them apart at this point but I and the other thing he pointed to the of destruction and so on so I'm interested in this because it tells me something about the springs that connected patents and of course that will tell me if I know this spring constant it basically as you know it nothing else then and elasticity constant if I use Koch slot it's up to the derivative of of a function like this but well let's have a look at that how it goes and so severe when I increase the exspy decrease the new why so it's negative here it's it's 0 so severe adherence to the end and here lies in this direction so when I increase the X by increase also increased by such boss so around this point here this function goes like this but so this is shown here so I have this is this this here because this is the potential is here you that this is the
15:08
reader relative to the potential incidents in EV per it's in the forest and when and to do now is to make the Mac nice and elegant I'm going to fit 8 spending to fit its sighing fit To this derivative missing this derivatives has pressed the shape here this eventually has to it because back to has ship rises to this part of me the derivative I'm going to set that signed through its 2nd the elegant mass and rather than any profound reason but again but I know this so and can calculate this derivative no problem so I I there is this particular point here where the derivative reaches a maximum that's at what about the 4 and a half years nanometers and it's at a distance of about . 3 compared to what I do now by then but sides the derivative of the interatomic that gives me this forests in the atomic we should always remember is the very very tiny distances right so we can linear arise whenever we feel it's it makes life easier so the distance From the equilibrium position In next minus L 0 is very small anyway so this function that we decide to use it is a mathematics and science the Delta next divided by 2 them the here and now we assume that this is that we stay very close to this point has been so we linear rights and we make it so but so that means we get this right and then we see we have a forest so we're going to have defined stressed that the problem with the defining stressed is here our I'd have to divide by certain surface and I choose in this case by when I don't have much training of strong arguments for doing this other than it's a reasonable we used elsewhere 0 0 square as the the area over which I will have that forced work so that gives me a stressed that stressed this forest divided by the surface area of standards and with this the flight use these 2 things together this is what I right now I'm going to the rearrange the formulas so that it looks like Hoechst law and hopes slot as you know like that stressed is modulus times threat With this I'm going to then say Well the strain is the difference I pull the atoms apart divided by the war regional some that's that's a good definition of the strain here and then all the other factors that pull to push put together and that that becomes my modulus right so this is the modulus then and and then I plug in all the data points that I have so have the f max In the previous slide pious pay of course L Is this larger distance .period 3 years which which came from the rolls out of the derivation and L 0 yes the this is a L 0 here this and this is at home but it's soon as I know the interatomic potential I have all these these things and and I can get you just from doing this from having interrupted what my elastic modulus will be spent for Irish dance and I find 101 the depressed 101 took on its is a little bit on this we will see that would see some data here this will bid on the low side it's reasonable but and but but obviously a little better simplified again so can remember we were working now we think about the lessons that we were weaving and the atomic scale here yet so I we'll just focus to make me make full use of it the advantages of simplifying the elasticity stressstrain curves by you making use of the symmetry of the crystal I'm just going to focus on the units itself a sec I have at the unitcell here and in some way or another and I have a normal stresses this direction this direction is directed and also have some of these red shear stresses there so that the question is now can I apply hoax along with the 1 you know from mechanics yes this is the law you familiar with from mechanics just cannot apply this to my single crystals and that the law you've learned in your undergrads engineering undergrad words which looks like this is and where where is deals the young Young's modulus and then this parameter here is Wessels ratio you it In the Sigma's and House of the and shear stresses and Epsilon and the Gamez himself and sheer strength that can you try this To this to this single crystal here would you can't you can't because many Christmas and definitely is not isotropic so you can pray in principle but I'll come back to this in a moment the principal you really can't use as you should be using this 1 this hoax slot and and so what if you if you have you compare these you see well because lurks a little bit similar except that the incident having here for 1 minus new on your new I was so I've got this fall see value and so the city values they are called the elastic constants of the crystal for the stiffness constants of the crystal and you can determine that that I'm just like
22:28
you do determined the all the elements in this matrix here by experiments to do that this the 1st before before it as something with it you can express the term just like you do for isotropic the now you can express the your hoax law for single crystals has stressed as well as a function of strains or you can express your strains as a function of stress and have the Matrix look similar nest but the parameters in them are different and there were a talk about elastic constants and stiffness here you'll talk about compliances and indeed there there dimensions whereas the seas are a pain Pascal of Pascal the ESA's foreign reciprocal Paska OK right so let's look at the person who was a few things about this how you determine decency values and the best values but if you know these their elastic constants C 1 wants someone to and for 4 years and there shown here but 230 stated 7 bigger Pascal's 141 bigger Pascal and 116 Giga Paszkowski I can easily calculate that it would be that compliances and if I know the compliance is of a crystal I can very easily calculate the elastic concept these are the equations the data given to us by alas this city theory single crystals and I went to the watches these quantities me a lot with knowhow and what should you do think about these quantities of so when we say c 1 1 1 How would you measure well it's related to if you have new single crystals yes and you stressed along the way the main axis along the cube like a 1 0 . 1 0 1 direction Italy 4 1 direction because of you will have a specific strain has had the relation between the stress and strain in that particular direction yes it's new this C 1 1 failure c 4 4 is related to to sharing of the same crystals so he sheer experiment and you look at the amount of sheer you get in a single crystal as the function of the the shear stress that you can determine c 4 4 C 1 2 is a little bit more complex to flesh out and there is 1 parameter that will be discussing a little bit more detail as as in In a few moments that is the compressibility or the inverse of the compressibility the bulk modulus so if I applied to this crystal a certain pressure yes you can actually make this crystal smaller and in general the reduced the lattice parameter and there is a relationship between the amount of pressure you give and the volume straight but in said this year the amount of pressure you give the volume strain the relation between these 2 is the ball much I personally know I always get a little bit confused when you know what's the bulk modulus has it's much easier 10 use compressibility yes that's the case value and that's that's actually indeed volume the volume the amount of volume share but and the volume of strength when you increase the pressure this volume strains over pressure increased that's the compressibility we like we like it to be a positive number so when we say miners with the minus it's just so that compressibility is a positive number because it by increased the pressure on this wound will decrease soul of wood indictment minus side its it becomes a positive number what such development is nothing else than the reciprocal of this compressible and what is interesting Is that mystery parameters that we've seen the C 1 1 C 1 to C 4 for these 3 the elastic constants which appear in In the hoping hooks lawful single crystals him I once you know the story yes you can determine all the other the bulk modulus compressibility muscle ratio whatever because those are the 3 only independent parameters of elasticity so it comes as no surprise that if it with that of compressibility can be expressed in terms of the modulus values 1 1 and you want again I'm not going to go into the last the theory here which would be just numbercrunching and seeing a moment that it's not worth the effort in the case of the so far of the the compressibility and of course if buy have the inverse get rid of C 1 1 plus 2 times C 1 2 divided by 3 for the bulk modulus right just a few things here
29:54
for those who were "quotation mark who may not know this but it's it's kind of interesting the I think to know when when you the for thinking about expansion or compression of the parent crystals Is that there is always the relations between the linear expansion of New steel irony the letters and its volume expansion and if we use so that we're looking at a situation like hydrostatic stress or thermal expansion yeah where you know where the loudest expense would you can show it it's very simple relation between the amount of volume strain you have and the amount of Lanier strain you get of if you the president unitcell here on the Sicilian itself and that's that's the 1 atom in plays and just let's assume we were looking at thermal expansion to make things says that any actions will move further away From the equilibrium position just a little bit the annual have a volume today but you will also have a change in the linear change in this direction for the very simple reason very simple relation between this volume change and be the a linear strain on changes that the volume strained it's worrying times the lean years straight and as a show of force simple equations that show you this the linear restraint is 1 over Al which would be won over the lactose parameter at times LTT that's assuming that were looking at thermal expansion in this case so that is the value of its thermal expense that's how you would measure it and the volume strain is 1 over the DVD and so if you look at the time if you replaced in the form of full of 4 beta you replaces the buyer Al the 3rd yes you find this derivative and then you can still that you volume expansion the value of the numerical values for times the later expansion of the crystal or if you compress the same thing would happen in oppression it would not right that well and lets them just you see there something about the the use of this equation was kind enough to put in an area where did this in what you do with these promises so let's say you have what we were looking at Compressor hydrographic compressive stress that and then we apply it to a single crystals of IRA and right and then we want to have a volume Change tend to just minus 6 volumes minors yes that's the volume Change we won and we know of a wiki elastic constants are a C 1 1 C 1 to C for 4 so so the question is will know what to how much too but should the pressure another 1 so of course we know are we looking at single crystal knows that we we know a lot about the strengths this and we also know that there is no shares in Greater stresses involved in this situation for them this is just a pure volume today there's no shape change so the entire I Take the a the axis for the stresses and strains along the Pew boxes so Epsilon 1 1 of whom little part equal yes and their their equal to minus 10 to the minus 6 here and so if I use this I find the stresses and strains Excuse me onethird of its value but it now applies simply applied this might my books law for me the hydrostatic compression so I have segments and here have my epsilon they're all the same and here I put in my C 1 once and C 1 to use them and so I this tells me what signal 1 1 as a single 1 wonders 230 times this plus 134 times this was 134 signs that's right and also segment 2 incidents and P is simply for using numerical values that's taking into account the change be the bigger Pascal's into Pascal's by writing down to the 3rd time make Pascal I find here 1 . 6 4 made of plastic all
35:43
alternatively but you have no alternative is you can use they concept the compressibility for bulk modulus too you get an idea of what to do of the relation will be between the reduction in the volume and the hydrostatic pressure so let's let's apply this summer just told you that the compressibility is not an independent elastic properties so we can calculate its value it's 3 divided by C 1 1 was to see 1 thing so if I put in these numbers this is the parameter by this is the value of the parameter and if I calculate the the bulk modulus it's a reciprocal of this and I find 173 the capacity to remember that number will use it in a little in contempt of the moment so what I'm saying White House much will the volume be if I have any that hit hydrostatic pressure of 5 Figure Pascal fault I applied here the equation for the compressibility and I find it 2 . 9 per cent and if I use the relation between volumes training and linear strain it also tells me that there there will be about a 1 per cent contraction in on scale right the 3 per cent contraction in volume a onepercent contraction on the linear scale well that's kind of interesting do people actually do these funny things I mean really seriously well yes they do there are people who actually take Irish and they increase the hydrostatic pressure and they go much further than the 1 board 5 Figure Pascal they go to huge amounts of hydrostatic pressure and so what happens when they hit a hydrostatic pressure changes as increased yes you pushing the atoms closer together this then all the crystal structures become more so if you reach at room temperature about 11 bigger Pascal considerable very considerable pressure you terror you L firing into hexagonal forms of fire called epsilon IRA and if you increase the temperature as for instance if you use your said about 900 deg C do the same thing you have at that time you have also like increase the pressure at about 13 the Pascal serious you turn your Austinite again in Epsilon Irish why would anybody do this who was interesting and well there's some people who were interested in our planetary science yes then you know what happens inside planets and site plans we have huge gravitational forces huge pressures and also extreme temperatures and it's very well possible that the earth which consists of the central ball 1 of the it's some kind of ferrous alloy probably ironical the and in those parts Of the In their structure where the pressure and the temperatures extremely high we actually have this form of Byron being stable and them not to do these experiments in very advanced experiments we have extremely high pressures and uses what's called Diamond and Vella tests where you going to do basically pressurized no sample Byron very high you can achieve very high pressures uses Oblast hands but you must also be able to study you so you must be able to to determine the crystal structure as you do this this extreme high pressure that way if you use the diamond in very high Heights strong tools and you can get very high hydrostatic pressure you can actually observe these transformations reason why you get you go from Alpha to epsilon is because the atoms are much closer to each other the atomic density is much higher in the world In the Epsilon the hexagonal structure and also the reason why I ate at a Austinite eventually becomes epsilon is also because the uh the density atomic density In epsilon is much higher the slightly higher to the Indian government but let's go let's come back to earth where you know I mean the topic of the course is actually steals so and then I think talking about the past the 3 quarters of an hour about about a single crystals of there are where can we say about that the elastic constants of the fire ants are well 1st of all there is a big temperature effect because the sort and with great there is a temperature affect your elastic properties change with temperature so there is different at room temperature and at minus 40 yen which is as you remember a temperature that cold but that's not it this To temperature and plus 50 in which it is warm but there are and there are many applications where we use deals with very much higher temperatures and power plants I know 5 600 deg C is not of no right to temperature is important and so you you Constance will be influenced by the temperature we never use pure irony most of our skills 95 per cent of Byron but it's not pure and the alloy Ewing has an impact the crystal structure has an impact some of our allies are for Riddick steals other alloys are Austin at Dickstein they don't have the same elastic properties you know that verite is Ferro magnetic at room temperature that it isn't at higher temperatures there is a small but noticeable effect of the ferromagnetic transformation on the much of this on the elastic properties I know we use Paula crystals we deformed them all the time that's what happens to the elastic modulus when you have a piece of steel that you may have the last 1 here than in the same piece of steel has been the form I knew you know what is the last mile is the same as well if you measure it yet but will now it's not a change because we have introduced crystal defects in particular when in the form of metals and steel you you generate lots of dislocations we already talked about Joseph generates .period defects like vacancies and then I very often also when we did for materials we create preferred orientations so is it words all the time 2 progress on
44:46
these single crystal properties it's just a question the 1st let's have a look at our the impact of the democratization temperature on the modulus so on the elastic constants the C 1 1 C 1 and C 4 4 I didn't change with temperature this is a very nice of you ever need to have an idea of change of your parameters as a function of temperature this is nice and her a relation between you and me 1 the value the CI Jay's at a certain temperature you values at 0 degree Calvin which were given here and then you knew of such track this parameter of the year temperature dependent parameter s divided by exponential team over the life of 1 the is the temperature at which you measure s parameter and team is also of parameters that did you get from this table that allows you to fits for instance the state here on the C C 1 1 as a function of the ratio of this is to seize the Curie temperature divided by obtained and you see those Cervantes the reduction of C 1 1 2 1 2 and 4 In front of the company of the equation I just showed you allows you to calculators that at any temperature were used but you see that all these parameters to soften them as you increase the temperature here and there you also see here that there is a little anomaly at T C T C being the Curie temperature that day the temperature below which the verite ferromagnetic and above which it's magnetic or dynamic nothing and you see that there is a reduction of the stiffness when we crossed the transition and in terms of C 1 1 but there is an increase in the modulus and deceit 1 2 of kind to spite effects but you should be aware of them if if you ever come across this year elastic properties are influenced by the composition new conceded here these are the parameters 4 Alpha Arun I needed the parameters for 3 stainless steels and the first one to focus on the first one because there have single Crystal values that you can see that I had I get different parameters and C 1 1 2 1 2 and for private and very good different but there is a difference between Gulf Arab then this stainlesssteel single crystal which is due to 2 things 8 the fact that we have another crystal structure and the fact that we have the other composition and will come in a moment about you know if you have to do calculations what kind of formulas can you use Texas 0 1 of the things you see in general is that if you look at the modulus hence the modulus and if you assume that your steel you you also irony is isotropic you see that the modulus of the Ferrari both the elastic modulus Young's modulus and the shear modulus is slightly higher in the In the 1st slightly stiffer lactose In the farright compared to the people the single crystals and parameters 1 wants 1 2 In a see for 4 allow you 2 the calculator all the other parameters yes in any direction were you ever to have to determine the elastic constants like the modulus Young's modulus of larger crystals congress for instance 1 of the things that is of interest to stress you want to know what is the modulus Young's modulus if I make a single an experiment with a single crystal has of Byron Nelson and I want to know what will be my model as well that I can use this equation here to calculate the modulus Young's modulus in any direction using again the story independent the elastic constants 1 1 the 1 who to work for and this parameter here fell 1 square mile 2 square of where helpfully square plus were allowed and what is that this is a parameter that it is but she owned a geometrical factor which which is related to the orientation in which you measure it yes and this is called the the yellow values here of the socalled direction cos so they're basically be the call sign of the on goal that that direction makes with the few axis of using quest so for instance it if we're looking at 1 1 0 directions and we want to have to know the the Young's modulus of this direction that's what 1st thing you do is you determine L 1 L 2 and 3 and you simple way to do this it's the call sign of the angle between the In the case of 1 1 1 1 1 0 in
51:37
direction is L 1 is to call sign of the other leading ones and 1 1 0 0 you know if I do this to the product Dr. product of these 2 I find that when this value and that is equal to the length of 1 0 0 factor and the length of the 1 1 0 factor of these ideas to square root terms times the cost side between those 2 direction so that everything is known here so that I can determine whether someone its "quotation mark Alpha and if you do this you find 1 over square room for a 1 1 of the square root of 2 and 0 3 and it can do the same here for 1 0 0 4 1 1 1 and you find these values but simply plug your see coefficients into this equations you see values into do the orientation of the value factor and you find the 1 1 1 to 100 and about he won 1 0 about 210 and the 1 0 about 140 conclusion of Irish crystal is highly an isotropic In terms of properties it's going to be very stiff in the 1 1 1 direction other it's that it's almost 100 figure Pascal last step in the 1 all direction so well From a principles point of view there's the the irony is not an isotropic material however yes again if we come back to steal and in practice they should be really really worried about that what we should be worried about this when it's necessary for instance when you're doing calculations at the level of dislocations or at the level of activists this you're trying to get some ideas of strains against in the crystal then you may want to worry about but at the level of the macro problem properties it's different so the single crystals that elastic stress strain relations for a variety of Austinite gamma you can basically ignored you can use hopes long sure Nice the isotropic solids now it's of course extremely interesting if you have a lot of knowledge about the single crystal to use this knowledge to calculate the average properties right and answer to calculate the average the monuments on an average shear modulus you using what you know about the single crystal problem because the reason why we can actually think of steels as being isotropic it's not because a single crystal is isotropic it's because of the randomness of your small crystals In the microscope too right that takes away this menace soap and there are procedures but to do this 1 of the procedures is to say Well I have all the crystals are random and when I strained over all the crystals strain in the same way and then you can do say this friendly you can say well now when I strained the stresses on the crystals are the same that's when you don't get the same results when you say all this all the grains are stressed the same way for all the grains are strained the same way because if you say there also strained the same way and the stresses are unequally distributed and if you say they're all stressed the same way then all the strains are distributed randomly at differently the interesting thing is that these are too limiting cases the right answer is somewhere in between so many of them into a wellknown problem averaging problem in In mechanics of crystal snares and so do we have to operating procedures 1 instance ISO strained procedure which is which also void uh procedure and the other 1 is the ice stress for the Rockies procedure that so what you do is you basically calculates 1 extreme to the other extreme and then you average you can use the arithmetic average or you can use a geometric ever for the arithmetic average ascended to an invited by the To or the every letter to metric averaging multiplied into a new take the square root now you going to ask me which 1 should I use I'll tell you you decide because there is no rule To do this this average and these are the equations so this is the the absence of friends and if you want to calculate the saying the mean average single crystals the shear modulus 4 Byron you use this formula but and again you see that a single crystal values here so this is the so strain value this is the ISO stress value you take these 2 values divide by 2 that gives you a very reasonable value for gene yes and
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I Of the because because you know the temperature dependence of C 1 1 C 1 2 and C 1 for you also know that that the temperature dependence of these average properties and there compositional defendants also good friends and Sears this is the mean average modulus don't modulus listening to every shear modulus for Alpha and this is devalued adopted line for single crystal 1 1 0 the the deed the but and it lets if I had the evil you of a material and the G value of amateur I can determine the bulk modulus has also took because everything is derived from these 3 independent parameters I can cover any other parameter that's and because I have the time temperature dependence and I'll Wallstreet animal materials and the composition of the kind that I can calculate quite accurately using these equations of the value is based on this is very close to the value we we had earlier is 100 and 58 the capacity this is for also siren this is for and Austin epic stainlesssteel to see the temperature dependence Young's modulus for the shear modulus and Debee value in this case began using this equation that we have from plasticity theory had against 100 and 54 it's very close again this in of posted slightly less stiff you get smaller the Italian 10 now
1:00:43
something really interesting about this the With parameters and it's a concept that's not a new concept of but it's an important concept for people that do calculations in particular density functional calculation and even though your elastic constants like the bulk modulus Walsall's ratio the C 1 1 C 1 2 3 4 4 lesser they are parameters of the B under former material that elastically deformity there's no plant that they did not related to plastic deformation right just related to straining the latter's will let but there are related in a subtle way too plastic plastic or plastic information in the sense that they relate to brittle or ductile fracture and that the reason is because the is related to hero right to steals but you can write materials here stills resistance to volume change 1 of the compressibility if if if the material is very difficult to compress yes it will have an impact on the compressibility and the the bulk modules to be is related to the resistance to volume Change to bond strength and fracture where is she is related the resistance of the material to sharing for defamation thank you the breeze and and and as a consequence the ratio of B over tea dance and we'll see in a moment and therefore Wessels ratio a lot very closely connected where the material is rental or ductile even tho these are elastic properties "quotation mark and there is another parameter you see 1 2 minus C 4 4 In it it's related to the type of bond if this value is negative the bonding is very direction is covalent if this parameter is the 1 who minus the Fort Worth is larger and until we have a more and metallic type of bonding non directional bomb these so that the idea of using these parameters to get you say something about the utility and so on the brittleness of the Crystal or over material that is based on its empirical is based on the study of many maturity and people find out love if the material is ductile it tends to be in this range the overdue value or C 1 2 minus the support for tends to be in this race but it works remarkably well and so well that when people do DFT calculations and they want to say something is material that the intrinsically ducked out of rental they'll say look at the elastic parameters we calculated by idea and also well we didn't do any plastic deformation we know nothing about the dislocations but because of these value the value of these brands we guess it'll probably be a ductile material and so what what what are these empirical rules the overseas smaller than 1 . 75 material tends to be right the over larger than 175 it tends to be duck so large and the values that means large resistance to so strong bonds but say that is what it is we're strong bonds a small chief values values he's easy defamation so the larger this factor is the better of the 2 small cheese large means In the case of Al do you retain fighter .period 1 this larger so we have you know we can protect you against it's going be what about the other part of the ductibility brittleness parameter that's based on elasticity is so 1 to minus the single for is negative and its brittle because of the the type of bonding I have seen 1 2 minus imports were larger than 0 it's stuff out firing you make difference it's plus 15 so material is just based on if we only 1 we didn't know nothing about IRA men with guns and the FT calculations would give us these these parameters we would have guessed well it's going to be mildly Dr. but certainly not break until no it so it's Let's go 1 1 step further so that and highlight y queso ratio is so fundamental that I wore a sense of restoration is is full material and you make the ratio of the 2
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cancel the thickness training over the longitudinal and and this parameter yes can only be we have a value between minus 1 and then plus a half and the reason why we know this is because again but was ratio is is the elastic properties so sold indeed it's been there all interconnected you only have to independent parameters so you can write there is a formula that you can write where this is very important for where the the question ratio is expressed as a function of the bulk modulus and the shear modulus so this is the original equation you can write it like that it's 3 times this racial minus stood divided by 6 times the same ratio cluster and so on while this value can be I was very large infinitely large size or can be 0 basic competence not negative so there in the 1st days of Wessels ratios plus 1 plus 1 plus a half Excuse me and in the other case it's minus 1 because in metals this is the righteous . 2 2 . 4 rates so so now we can reuse this idea here is to say well material brittle when the the ratio has a certain value what this of course has because the you've achieved the story that was so ratio is is this that has this dependence Of the bureau chief ratio we now see that the deed the day was orations actually fundamental parameters for a brittleness to let in Crystal so and supplying these 2 concepts to all of our so here you have the let's say here 1st and this is the view over G ratio and this is the difference between C 1 to C 4 so we yeah the material is brittle if you look at the White parameter here is negative and the text parameter is smaller than 1 . 57 so this is a man any material that sits in this corner is brittle the AlFaran is here yes it's in doubling the ductile the area indicates that Of the you can represent the same and the lady data in this G verses of the graph of Europe G values the values lines of constant the overdue ratio are straight lines in this year's and of course if I have is that the constant of value I have a constant the Wessel ratio right so so here it is the overdue 1 . 75 that corresponds to a queso ratio of about 0 . 26 if I'm below here yes be over tea is smaller than the 1 in the brothel area that's that's look at what where alpha irony will we have achieved that's about 80 remember and that's around 150 In some right here this very slightly above the brittle area To purely on the basis of the year of the year an elastic properties we would conclude that the science and steals art and we expect them to be ductile but not extremely doctor what this approach the sentence that something fundamental about you know what to expect but because it doesn't say anything about this locations which is ultimately responsible for plastic information and it doesn't tell us anything about strain hardening yes it tells us something fundamental but it doesn't tell us the whole picture we do need to look at plasticity of crystal hellfire and endemic crystals to actually know How how brittle or how the the ductile materials having said that but I didn't put this in a this Kraft because I'm the 1 he true to my promise that we only talk about iron and steel in this quarter but if you would put on the same bracket with other materials you wouldn't find that materials like Silicon very strong material but very brittle and all the semiconductors are here there's a lot of oxides are here that very soft metals like gold and silver are slightly above the fire and etc. so it it does it works remarkably well these things predictors the plasticity of order rather brittle ductile behavior in practice and
1:13:08
now just to finish In the material you will get In the end there are also a number of them Nice equations that you can use To compute shear modulus when you do calculation With dislocations this because of this the cases are related to shape change the Maine parameters that it playing a role yes here is the shear modulus so you kind of you interested in knowing what is the shear modulus both gamma irony in and the temperature dependence and their dependents composition and so these are a very useful equations we have devalued for Alfred and then the program as a function of temperature newly last awhile I don't really know they need survive well because because I can measure it right it's a like a measure because they can do with test and I can you measure I my judgment on going the ghost money just just a case something from fail is useless you measure your elastic modulus and you measure your was oration can do this in a routine test right and you'd be right however as I said in my introduction in many cases the findings in steals the phase there you want to analyze the get the mechanical properties from friends and if you have a complex my perspective that particular phase cannot be made microscopically the In many situations where In steals where phases are stabilized by the fact that they're very small and in that case you know if you want to protect the stressstrain curve of you composite you will need To have the shear modulus Of that phase that you cannot measure in practice and it's very useful to have equation like this Of course this is for pure iron and then if you want to get the compositional effects fall there you have it this gives you the change of the modulus of and the change of the modulus indicators of Austinite when it's Allen for you In reality composition certain elements will increased increase the modulus other elements will decrease and the fact is you can see that in the late in their the effect is wrongly similar but not all this the sequences here there are elements such as Crohn's which have a positive effect and also in variety and negative effect in Boston but you can use this time around stove information on these 2 flights allow you to basically get the shear modulus the average shear modulus of a face as a function of temperature and 4 different compensation but it up thank you very much to for good will but overturned the you must do to
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finish here and there and so will meet on on Monday but so I might as I said I put in the year that the information the slides from the Bclass 3rd stage let me know if you can access them