Mechanical properties of steel 6: yield criteria for plasticity

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Mechanical properties of steel 6: yield criteria for plasticity
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The sixth 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 plasticity, yield criteria, multiaxial stresses, Mohr's circles and R-values.
Keywords The Graduate Institute of Ferrous Technology (GIFT)
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but after a week of vacation let's try to reset our minds to new area class spirits here and so but with the we got to last week was the the concept of a yield surface and so did the ideas Istat on In very 6 something like the stress space and if you want and done to any stress you apply to any material that can be decomposed in some hydrostatic component and the DVU Tory component right and of course problem is always like so why not because you know what do I do with something like that well it's it's what does that mean for incident practice when you to attend salt asked or any test of you know that this is just rule applies so let's let's have would you repeat with which said last week about all business concept for a tensile test right but the tensile test I the euro and we don't have to make a lot of analysis we know that the the axis along which we stressed the material I will be a principal axes and so we call this France's Swan and we applied a certain stress along this X 1 and and if we pull hard enough so would if the signal 1 here is this large enough reaches the yield strength yes and I will have I yield knows we're all familiar with that it's just what we do when we might have frustrated but before that happens yes under let let's look at this and this distressed situation having the material right what we can always composed it in the a hydrostatic stress or deviant or express this this basically means that but I can decompose this the vector here yes in stress plays no signal 1 sigma too signatory being principal stress directions I can't that decomposes in 3 components which equal size all along X 1 X 2 X 3 and that's just the mean value of the story so so I mean value if I only had 1 stresses that stress divided by 3 right so I do 1 3rd of X the Sigma acts in 1 direction one-third of that the same lines in the x-direction in this case and in the White direction right so that the sum of these 3 vectors equaled actors is the hydrostatic component of distressed that I have blocked it so you made not think of that as you know when you do a stress-strain curve that there is a hydrostatic component but actually you can always decompose distress you apply in this is hydrostatic component plus a deviant Torah component and ended this case deviate component is what's left yes for you to get the and to get to this specter so that's a defective going from a year or 2 the this In case and that it so this an end to this vector viewed the components of this factor as it is of a R the the Eutaw extracts of and and very easy to determine what I'm here I've gone 1 3rd in the direction I have to go so I have to do an extra 2 3rd here yes and for the other to adjust have been negative values minors 1 3rd of the Sigma exon minus 1 3rd signal right so that gets me back to where I am I was that it is doing things this way I it is interesting because we will will will go down to the work we arrived too India do do disgusted to fund these criteria for yielding is set this up uh hydrostatic stress and end this deviant Torex stressed yes the perpendicular to each other the perpendicular to each other always nothing the Collyer to conservative of you can do the more general the revelation .period the 1 that you have a your like it you always get this this nice rule so and
now we go to where the we come to this witness here the last time and we we
have seen that on the basis of by of work idea yes distortion work idea that you could come to this phone meets this criteria and so the theories and the slides that would we want to they arrive to assist this it ended up being dismissed really nice symmetrical equation Nos and does so if you look at it in From a geometrical point of view pops of there you
find it's nice that Jupiter that we call the but the police it's the surface sealed surface now that before we go on in the 2 parameters that you sometimes come across yes and have already talked about 1 of them that's the equivalent for effective stressed so it that the term here on the on the on the left right that that term here we call the effective stress and this is square of the effective stress and so and so the equivalent or effective stresses the square root of this OK and is that you don't have the slightest interest made it but they should be gone up so the yield criteria if he wanted to express it in a very short this possible way is by saying Well the equivalent stressed reaches the yield stress and on the basis of the this equivalent stressed definition you can also and defining the equivalents to rein yes and the and of if you knew you did the derivation for the equation for the equivalent storing or strength is by defining by defining it by psychic the it's incremental work per unit volume you can define it in terms of the equivalent stressed and the equivalent strength in this formula here In terms of interest increments of equivalent and so that's this is familiar equation for the the work per unit volume of of defamation and insult if you if he were not going to go into this because it's another textbooks and it's not a big focus of the course at this stage but after some manipulation you'll find this simple equation and it is this useful to have this equation and will that in order to have something like an equivalent of a strange will come to it in a moment when we talk about forming and that the properties material properly properties that you get from forming tests it's nice to have an equivalent of strength top but then so that's come back now to this this particular equation here this particular equation here so the geometrical forms this aid is based media the equation Of the cylinder surface the diagonally oriented In the stress space where the axis far along the principal direction and now Let's Make the Connection now with which we just said about a hydrostatic stresses and Didier torrid stresses so that any stress yes Sir Francis this point here yes but you can do that you can do equivalent of construction has any stress units 2 of the stress the situation would be from the origin to point be yes that defines 3 principal stresses of its distressed state of your material has you can always decompose it in a vector going from the origin to a plus from a to B. yes and these 2 factors are perpendicular to each other and the first one From and so decomposed that this figure here what happens here into 2 2 bets the 1st but here is to show you what is the the hydrostatic part but the hydrostatic part we take the mean value of distress yes the distressed state and if you do that that is a so the mean stresses Sigma saying yes so you have Sigmund in the x-direction Sigma in the wider erection and Sigmund and the Sea direction so the length of this factor here is square root 3 of the segment look at not so about studio the hydrostatic components and then beat the Factor 8 to be here this is the square root of the few Torit component of the stress along the insurrection y direction and see them right until the end of course that the length of the factor from from the 0 2 B is basically the length of the stress factor writers the square root of Sigma X squares should be supported by the way I was signal while pleasantly the please make the squares the but if so right to so we have at the deviant stress and the the hydrostatic stress and and this plane here uh going through this point here is the deviator play right so now if I want 1 of the things that that is of interest in particular force force many steel products and particularly particular flat steel products is the case where but we're interested in the yielding of the material in a situation where we have the plane's stress the in a situation where we are we don't have a stress being 1 of the the principal director segment free and it's 0 so so that then we're basically looking at the intersection of this cylinder when the X and Y plane yes and when you look at this intersection you find it's this you find this this curve here which looks like an ellipse nest which has physics equations and you
can derive it just south of is very simple you just you just would here the Cigna's equals 0 that's it's not that difficult to find us in this equation here is basically the same as the 1 users equation and is so so what would With this basically tells you if you have a sheet material I you pull in 2 directions now in in 2 major principal directions when you have a stress states that there is on this point the materials should yield yes if you should see yielding and so you can do this experiment but and there is a machine in the In the big Paul yes there from the animal group which actually does this unit it pulls the material into perpendicular directions so you know that you have that's the direction Sigma X where you have stressed the you apply another stressed in the other direction yes when you look at you 1 wonders material start to yield In and Out so for instance this point here would be the only pulling in 1 direction is like doing it turns out right and there if I go along the diagonal here yes this point here is I'm pulling I'm applying the same stress in both directions and light then and that the nite and note at what stressed the material starts to yield and that gives me that particular point yes that particular point and if I if I do different over stress ratios are increased the stress a different stress ratios by Icahn basically described b a segment of this yield curve yes and then you can do this for 4 steals movement in Europe different skills as is the highest-ranked low alloy steel drawing steal a very formidable I steel end of that and so the the the softest material has the lowest yield strength so the smallest Of the ellipses and as I increase the strength of the material the Ellipse becomes larger and it to it's like anymore more stresses larger stresses to initiate yield of 4 that's 1 thing the other thing is that I this yields surface according to fund these this no on the seems to work pretty wet having said that US if it doesn't always have to work but and there's no reason why it has to work out the many things that can change the yield surface and and and in particular 1 of the things that there it changes the yield surface and I saw trapeze and I saw repeating In particular the normal anisotropy and playing the anisotropy when we thinking about cheap material the Inc you get another he'll surface on the yields and then the so for instance and let let's go through the the different bosses simple possibilities you can eat your material can be off the proper has been in and think about it basically most cold-rolled sheet Bob which we call off the property that means I you you measure differences are values in the length direction and in the transverse direction to roll and board these materials beat the yield the equation looks like this but it has the basic 3 terms is Sigma X where 6 of the product of Maddox and signal widespread equal to square of the yields were right but it has factors additional factors which are functions of the planar anisotropy if we have only the normal anisotropy so that means that the are of value in the directions parallel to the rolling in normal to the role of the same and equal to are then you just there Our 0 an hour 90 equal to each other and this formula becomes this 1 does and if you will we don't have any anisotropy like the op and then the ah value is 1 of the that then are 1 but in the articles that are you acquired 90 is equal to 1 we got back to our original for me this equation you can help but in most cases there's the materials and in particular steals because we deformed yes 2 to shape them into she 4 wires for being so yes there will be any measure of anisotropy and the reasons why is because we get when we when we do this defamation we introduced texture crystallographic texture and that's so let's look at some of the material here just as an example of this is immaterial as a forensic 1 of the stainless steel it's 400 39 is fighting and stabilize its it's got a very very boring like rastructure you can see here the Microsoft rupture just the far-right drains basically and I an end to this material contains 17 per cent of gross that never goes through any transformations of self a very simple material you can already see but if you look at the grain structures in the normal direction in the rolling direction transfers and that there is a slight difference but In instructor this right the differences become even clearer when you are doing you
taking stress-strain curves that I hear we have stressed the blue the black 1 stress-strain curve at 0 the red 1 at 45 degrees and the 1 in 90 this is the blue 1 so you say different the measures Of the mechanical properties depending on the direction in which you did the measurement that that's they know what engineers call anisotropy plus L which would stop here in this article slide we have changed the amount of defamation here it says that about 70 per cent of the rolling reduction and materialism yield nature 75 80 85 per cent about of rolling there and you can see that the anisotropy gradually decreases so I again have just so you know what this this illustrates is that if when we already know that the only real strength the tensile strength is not a material constant yes but because of anisotropy even if you being given yield threat or a tensile stress you also need to know whether it was taken in the normal direction or in the transverse direction to rolling direction like in this case because they can be sizable differences and and in particular the material may look batter yes if you do the measurements in the transverse direction than in the longitudinal direct rights it's always important I'm certainly 1 technical discussions about and there's only a single values being reported unique you should inquire about in what direction it was made and you can see here the values How the values change as a function of the uncle and that the function of the other and the amount of deformation of the long day will change the prince's using the total elongate of you know can be as low as 30 per cent and as high as No 36 per cent the the anisotropy is this real this is this is real .period and of course we were talking about the aspects of the U R value the normal anisotropy mind you want the art value this is the ratio when you make a tensile test the ratio of the In the with over the thickness strain against you see a very large differences in that it can be the at 90 degrees it can be it over to yes at 45 it can be at no less than 1 and that so and if you calculate the mean value it increases wait increasing rolling reduction and the Delta are the playing the anisotropy it decreases with increasing role in the that is the reason why when you want to make an early formal part she material are you will want to developed a very strong texture yes the very strong affection which gives you know the plane arrived repeating a little or reduce differences the the properties in in the plane's nose and a high art value and and this also shows you that area you know the same material say France is the degree material here it's very important to know what are value is being given to you because you can have an R value as as high as 2 and united and 1 . 2 in the same material there is nothing wrong with that but you do have research and data out there where you know people will claim or have our values too I have are valued Japanese a really good at that and that our values of 2 . 3 they don't tell you how its measures very likely they had measured it at 45 degrees to that beautiful data but they would have been very low nothing abnormal with that it's that it's very the important thing to be aware of the fact that you do have this displaying their financial look at and and so the reason why this this this happens is because of you developed and in particular for steals before it exudes a new development is called the a damaged fibers and so at the end of the year texture Chris study of crystallographic texture uses nowadays in the past we used used to use poll figures to illustrate the development effective nowadays we use orientation distribution for France because they allow us to evaluate the presence and intensity of the Of of many texture component to see whether this is the result here of such a measurement there is a black line here means that there is a high density of these orientations and if you look at this time the schematic here for the corresponds to this particular would you have what call section you see that they all start all the components stocked with 1 1 1 plane so that means that all the the grains most of the Greens many of the grain large volume fraction of the the Greens have the 1 1 1 plane parallel to the surface of dish that and maybe on that's very good for form ability because it leads to high are value and not only high are value but if if I may go back to the lower delta our value is seated in the blue case the delta is reduced also that's that's positive direction and
and and so you can you look at this this whole science of texture development in materials science and in still development has benefited from that you see that you can have you can look at the intensity of different components texture components as you do the deformation so more and more and more information you can see the texture component with very little R value decreases in intensity picture components with high here there are values and the ones that are very high 5 5 for his very close to 1 1 1 on it is that you can see the increasing very much wants you do additional information so that's the reason why you get you know how you
control the the detection of the best this was an example just that for a a stainless-steel a frantic stainless steel you also have it for a regular non-stainless sporadic steel for instance I have steel so if you measure your at different understood the rolling direction you always see this very familiar and I stopped trapeze but the highest are valued at 90 degrees and that's usually the 1 that producers but will be recorded yes so if if if we have these values at different articles on you can derive as I said I mean value of art which is given by this equation and the that means value of Delta these are the the 2 equations so if I have values the 3 measurements 1 4 or 0 . 45 and our 90 I can calculate the mean R M B and the Delta so we are in values or better than we didn't just part 90 because usually that's the 1 day you will be given if you don't ask for anything else now OK so we we would just show there the aura of value has an impact on the yield surface and let's have a look at what happens what we're going to look at the simple situation where and there is a normal anisotropy to the B R value In the plane Is the size thus and such events which which is seen years the yield function looked like this and so if we I I take this equation now and so I divide by the yield strength here there's no strength square and I can rewrite this equation like this and I'm very often people like to we write it this way because that makes it very easy to love and to show there are a few things related to plasticity and anisotropy so if R is equal to 1 we have to the normal no for says lips again you union actual tensile stress would be along this line of by Axel tensions will be along this line and you can also have portion of simple shear or also deep drawing is similar it's a long way this 1 the these under the tentative which took went when is equal to 1 no we know that we have an alleged is equal to 2 the elapsed His stretched out it's kind of interesting and I do want to .period some important features here so this is what happens when when are In creases so again when you have when you doing you union actual tensile all the stressed yes you basically start from 0 L I knew increased stress elastic elastic lasted less than deal you had the yields yes the yield stress and before like that could also you is a situation where you have across like specimen and you increase grade segment Ex and the signal y With the same amount in the same year when you increase the Delta wide by a certain amount increased at the Sigma X with the same amount has led Europe in Europe you're stressed state goes along this diagonal right so if our value is 1 it will it'll start yielding in this condition if are value is too it doesn't heal because I haven't reached the yield function so I have to increase the stresses so but I think there's no chemistry involved here is there's no defamation involved from here to here the material as elastic from here to here the materialism last nite so but the fact that you have a higher our value means that basically it's gotten stronger for this particular stress state so and because this you get higher our values because of that crystallographic texture you you call this effect textures strengthening you basically texture strengthened the material it doesn't matter how much you texture strengthened the material this point immune the actual In the unity actual can test there is no strength and it is matter how how textured your material as it always yield at the same time that's what this is about this tells you so yielding just as important as this is a function of the stress states this very clear illustration of that and um and anisotropy influences the the the yield .period with no conditions stresses at which you get killed another let's go into the fastest real plasticity actually because what about 2 now when we were talking about the the yield situations we never really talked about plasticity which is said well if we go up to this point you know the material Billy elastically before them and it's only when we reach a certain amount of distortion energy yes yes that is equal to the same distortion energy it takes for a uniform so tensile test to start yielding then we have yielded yes but which doesn't really so what happens afterward you know what so well From there 3 basic cases that people of described in plasticity Is that let's have now that connects our surface with the stress-strain cover them so this let's look at the familiar stress and then connected with this yields to let's say we are out we apply a stress among the Y axis for a change because it's more convenient stress-strain curve so I increased the stress and strain increases yes but it's all elastic right and say we have what is called a perfectly plastic material was perfectly plastic material doesn't dozens strain hardening right this well as soon as you have reached the the yield stress the stresses remain the same if you want to continue deforming the material and so you have a flat the flat plastic part of because OK now I reversed they the stress a reverse distressing we have gone from here to here the yielding yes is at this point I think continues training the material this will remain the yield surface at all times the matter how much former material because this is what the stress-strain curve looks like I go back go into negative territory yes material yields when I reached this point that's exactly the the miners value this field strength against a dozen pardon and the yield surfaced basically remains the yields and that's the case very simple case
of no hard and Of course we know that that's not the case in many of knowing when we have a stress-strain curve of steel it to look like this have an elastic part the material will yield and then I will well and you know if I strain of a little bit I'll end up having some kind of flown stressed yes so it basically means that the yield surface expects and in this particular case expands uniformly yeah but when it expands uniformly like drawn here we it's isotropic part that you now I want to continue the story here now so my heels after I performed at the bit Beyond that the yield stressed beyond elastic limit for years and this is the now the new yields surface yes within that lets go back years and I do the reverse I go In reduce the the the applied stress and I go into negative territory and so this now tells me that the material will yield not at this point not when I reached this point narrow the material now yields when I reach this spot good so every time I go back and forth yes every time I go back and forth do you know I can go back and forth like this distressed finger from going to describe will you become larger than but it turns out and in particular that's it's the case with which steel very often is that that doesn't happen in practice In practice we very often don't have this isotropic hardened yes and I'm sure that you have heard undergraduate classes about the so-called boasting the effect abolishing the effect consists of the following when you take a sample steel sample can you pull it as a measure of a certain amount of yield stressed so you think if I'd reverse my the direction of the status of attention I go into compression I should get the yielding at the higher value then the yield strength here yes right now because they the my yield surface has expanded it turns out that that's not the case your yield strength is yes that's true this phenomenon is called the motion Arafat and so how do you represent us in the the and how you still use the the concept of the yield surface right on it if you can't trust you know the direction of the stressed that the new replied even in this simple case like a tensile said that as well we say that in that case we have to nomadic hardening yes and instead of expanding of having it isotropic expansion of the yield surface we the translation of the yields surface in stress things so the when we start we apply a stress we passed the yield stress the material now hardens as and I know and I reach a certain flow stress and I stopped my task so now you know if if I were to repeat the test the decrease stress and then start again this is where it would yield right so much I know this is now my for me surface yes or no condition that if I move my yields function in the direction of the applied stress like have long hair and Goldblatt units and now it should normally yield at this point because I have shifted indeed deal surface upward it will yield here at the lower value against and this this allows me against him 2 described plastic behavior yes if boasting the effect of occurs yes and in again so In the case of Steele's yes we know we will Bush in is rather common occurrence for instance I in particular in certain industries it's really important and other industry it's not important obviously it's related to the fact that In service business deal it is still subject to compression for attention car body that is rarely the subjected to compression because if you buckle thin material it will elastically collapsed before you you get into compression you have Buckley which called Buckley has In the case of however of thick walls pipes yes it's devotion reflected import of what forgings for instance look at so so when we change but when we change when we mean when we say stress the material beyond the yield points which we go into the realm of plasticity and that in the realm of plasticity there's no such thing as the hooks slots doesn't exist right there is no linear relationship between stresses and strains yes but the race there were some fundamental relationship and that before we start with fight and we we need to say something about 80 the deal surfaced surfaces something in stressed based prepared but it's got some very essential bits of information about the plastic strain increment yes so when I change stress yes there will be straight with information workers I'm in plastic region and of the deformation the well the young surface tells me In what direction the defamation will go right people used to be clear in a moment but let's just say you accept this from me right I'm so so what is it basically meet what with what is this about this essential bits of information about the plastic strain increments it tells me this there that the strain increment is always perpendicular to the yield surface the of of this this is the yield surface somehow so I draw the tensions To me the yield surfaces and somehow the factor that's perpendicular to the suspension yes tells me something about the direction of the strain encourage them and this is a very important rules governing the normality will yes NBC says that the plastic strain vector is normal to the yields of America but let's just have a look at this the latest assumed we accept that yes and I and we look at what this means in practice talking to this is the simple 4 of the equation for Article 1 no anisotropy and I can rewrite it as fast is equal to 0 this is like a function right which is simple mathematical function
as of these 2 variables is you look at it let me calculated what is the but uh defector normal to the yield surface but that if I want to have that know what effect arrests just to the derivative and derivative to which respect to Sigma X and derivative with respect to signal y and the sole derivative is here well is two-time Sigma X minus signal y and then the derivative with respect to signal wine is minor Sigma X plus 2 times signal it is a derivative the any any place we believe this is staffed the derivatives to ask has 2 components listen to anywhere here yes this is the coordinates of the derivative to that equation the 2 after but and let's just have a look at a guy we were just having at this point right so in this .period segment X His equal signal what right so I think I put this knowledge this is so characteristic of this particular stress state invests who Sigma accessing the wives this is Sigma X ,comma stigma what with so what is it tells me is that they do normal here is of course .period in the same direction as the as the stress yeah the stress fracture that knew to Satan while you know Hi this is very simple black writers start we knew that it didn't have to do the calculation right because this is a it's a curve that way and can so but amazingly means you hear my mind the formality role tells me that if I notifies trainees directions that I will get a component of strain in the X and Y direction which are equal right after I saw nothing really special now OK well what happens if we do they are the union actual tensile right should be even simple right are there should be 1 component just to stress that the strain desk in the direction of distressed right that's what you would think the of course not there isn't yes yes now but it took the 1st let's let's look back at the simple case which we which we just had where we calculated derivative on the With the police's equation for Article 1 and now we're here but if I look carefully this is the attention to this line has been the normal is not parallel to distress it's not like you can see the whereas I only have 1 component index component is for the stress of actor I have 2 components have an X component and NY components for the students trained for the strain not all have a tax component and white comport and the and so if R is equal to 1 I calculate that as used in this way and in this case but if the signal light is 0 we I have to say ,comma X -minus Sigma effects this would be best of new tells me that the strain in the X component will be twice the strain In the white component and the and industry in the wine direction will be negative not positive when you come to think about it of course when you pull a material yes In the Line direction yes so it's a flat material say it's not standing in the way the direction right just just so wouldn't call it's got it's got to contract right so there are indeed too principal strains yes for that particular stressed state in Europe tensile specimens but it's so so now out of we can we can really do this this type of calculation using this equation where are is not equal to 1 but it's equal to 2 and in this case it turns out that Ferrari closer to we get the the have the segment two-time Sigma X and the other 1 is minus for 3rd segment Texas so when are increases yes went up to the plate however figure that's when are increases these the West Over length ratio the strange changes this is shown here but this normality rule means for us to so in the case of by actual the stress and so we go along the diagonal here it means that I get to strains in the positive Sigma X and signal y direction yes a deed you know the actual stressed directions that is what happens when you do a tensile test I have a strained when I pulled by strain not only In the accident but also underwent directly material good becomes standard answer to this is this In that the material becomes less why Excuse me what and and this is influenced by the ah values so this is a show here in in the schematic here so that this is the yield surface is the straight line as the attention to the surface and this is the case for Article 1 the ratio of the wife over the X yes this is a long the the tensile direction this is perpendicular to it is one-half minus 1 if I increase our from 1 to 2 same equations now the with but Over but this is an area that the if you have this so this should be a wide divided by epsilon X which just noticed please correct because it's an error here and here is that this would be the epsilon X but the drawing a spot and here it's so it's a case so it means that the do there you got you get more went straight With strange increases from you because of this ratio goes from . 5 2 . 6 2 write to the ah value also influences the strains that we get but if ah increases you get an increase in the wet strain and as a consequence a reduction in the thickness straight and that's really nice because it means that you training material in 1 direction and it will tend to get narrower the unless white yes the larger the are evaluated this shows hero that you get less then because it went strain becomes larger but look at them before we go
back here but before we get this mother of interesting situation as this 1 here yet because you may wonder fired their situations where we only have strained in in 1 direction yes and there is there is if you look carefully at your yield function there is this point here yes where the tensions is vertical yes and where the normal factor which the the role of normal to says the this vector is In the dot that To the increment in strain is parallel to this this normal as you can see that here I only have this strange In x-direction yes yes and it's not it's not the tensile test right it's not a tensile test because this stress state this here this is stress this Council the statement faxed to the wine industry so it's a test you would do for instance could do here so what happens in this case you missed this case is called yes history is called playing strained Case why is it that because the epsilon why this strain In the y direction is 0 yes so the material gets deformed in such a way the exact yes there is no which strain so but you know that when you do plastic deformation you the volume has to remain constant the few make it longer this way has to defend him so plane strain it's a very dangerous defamation situation because any strain in the major direction is automatically compensated bye right and when materials things you get closer to necking fracture right so this is the answer planes strain conditions are therefore always I'm very bothers him In many situations but it in informing also have complained strange situation know how does this work there but obviously remember when you're the reason why aim when you do uniform Union actual tensile test the these the material on the sides is free to move it's freedom of yes when you doing it plane strain condition you basically prevent this from happening by applying positive stress on the on the phone on the sides this that's the reason why you need to have a positive signal y yes to prevent Al deformation in the with direction the but so I right to know we we've we've got said a few things about yields surface I already told you that something that must happen to deal surface as we strained the material we did and it can be no hardening it can be isotropic Lee hardened war if I have things like the abolishing effect the unit kinematics Cardinal the shifting of the the yield surface fame and then we have is really nice role gas which is normality room which tells us something about the directions of increments small increments of strength so and think plasticity what would the people work which is not with stress and strain but which stressed that increments of string right so do we need for 4 we need hoax lost because we know we need to we need those for the initial stages of the committee and we need to know when do when this visits staff to apply these elastic relations laws we need a criterion for yielding to determine at what plastic deformation or what stressed that that would stress plastic deformation spots then we need a work hardening rules yes and so we need to know how even in the isotropic state we need to know how the the the yield surface expense and we need to have flow role and this is the last thing will be discussing which describes the evolution of the plastic strain and it basically that floral it's it makes it's basically the equivalent if you want to coax law In elasticity but it really doesn't really stress 2 defamation it relates to increments of plastic a small small amounts of information I so we just talked about the nor normality and I told you the the this strains this incremental strains are related to the yield surface In a very simple way yes Is that the thing this string components are the components of the vector normal to the yield surface that's very nice right so if I if I know the derivative of of this curve and and very very simple and it's not simple because I know it's this guy made a very simple for this course since that is that some of them is proportional to the epsilon X derivative to y y directions proportional to the at lower in the absence of you the trouble is of course you know I I I only know but something about the fact that the proportional doesn't say anything about the site's yes and and that's why we need this flow role relating increments these increments to this increment to this stress yeah then again so the funeral yes the so we will just again because it is not really necessary to make the things that complicated at this stage in the course of Will will just look at the Union actual consultants and look at the full rule for this the Pichler test and you can you know you can expand into but the general situation actually the equations don't look much more difficult so it's so this is this decision needed that this is stress state you know the same snobs 0 uncivilized here 0 in art you the actual tensile test we can easily calculate the mean stress and the deviant or express means stress is one-third of the sum of these 3 guys and so because only 1 is non-zero it's segment is divided by 3 and I can also calculate the expresses so the so that's a prince for Sigma X -dash it's segment X minus the mean value right so that its two-thirds of segment expanded signal widens minus 1 3rd of segment expensive machine is one-third and then I can also I look at the time so these are all stress related equations right this now look at the prison created press ability now that we have in the case of plastic deformation so any again if I stress in 1 direction plastic stresses in distinctive because stress in 1 direction that strains in the 3 directions that must be such that when you some them the some 0 so the sum of the X the Wise the the epsilon X the X 1 1 in the depths of the sea is 0 yes and uh and and obviously I'm pulling in 1 direction so the Epsilon this time the reason why you find this site you go from here to here maybe you want to add this on your slide that is because the epsilon next year the epsilon why Andy at Swansea are equal to each other right and then you will find this equation right and and so on if it you know combining these the these equations and these equations that you make ratios so you make the ratio of the deviant torrents Charest component in the wider action expressed components x-direction make this ratio you find minus a half if you make the same ratios for but not for the strain components the UPS said long wide over the at symbol next you also find like 1 right and so you can combine these 2 equations to To find something that's really interesting namely that the Strange increment in the x-direction divided by the deviant horror stress component In the x-direction is equal to the same ratio in the wider action saying ratio in this indirect yes is basically tells me that this ratio somehow is constant and is a constant and in what we do and we'll see what we call this we have a deal under which is the name right just a name and we'll see if there is any way to determine this do you like them this but right and basically we're there we're there because I now have a relations between but my plastic increments and distress the exports is equal to the lender which is kind of a the obviously it's sad when you look deeply think deeply and it's not much more intelligent right because in another question is what's the land or so anyway we will see in a moment how we do this but people like to but sometimes change this equation into an equation where we don't have the view torrid stresses and so they will change the he admitted will change this deviant Oryx stress here into In terms of the the segment Texas signal accusing the definition yes To this allows you to now too we write this equation of these equations "quotation mark for D that the capsule next year to avoid the use of this disease In this matter very very symmetrical but and and eventually 4 union actual tensile test where signal wide Cigna's yards roll it gets simplified the decks and so foreign of test Hi this is what I get I get the segment of the natural next over cigarette taxes two-thirds the London as and have a nice relationship between my increment of strain and stress because to so the question is of course what does it all I mean that as well as as in the case of a the tensile uniform tensile test if it's relatively simple to illustrate how you would how you would get to the land so you know that in a uniform in a in a tensile task I can replace strains stresses by their equivalent values has basically get this and so the stress-strain curve you have stressed curve here so which it is you stress-strain curve or you can write it down as an equivalent stressed equivalent strain curve yes it's the same In the case of the actual test so but below the yield strength things go along the elastic modulus 10 of them beyond that Our 1st train consists of an elastic part and the plastic boxes to constrain looking up I can take out and use my wrong data gear to take out only the plastic strain so functional starts at the yield stress the increases as well we basically look at the specific the yield this is be flow stressed equivalent stress here and we look at a small amount of the strength of the Epsilon here well the curve now tells me what's the land s if this is the epsilon this on goal here being tutored the 1 over detention of today is the ratio of this divided by this the president epsilon divided bicycling is nothing else than two-thirds all of this consider the gives me the Nederlander I can only do this of course if I know that I need to know this behavior and so this brings me to the point is that you cannot get this the lender on last you have some way 2 if you have some theory of strength in Europe steel and unless you have a theory 4 the hardening of the steel this if you don't have that you don't get to know you won't get this the lender yes and so that's what we'll be talking about there in the Didier In the coming lectures about this theory of of strength and of hardening of steel which allows you to apply the fastest focus from a little bit over time because I wanted to end of year thank you for your patience and


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