Mechanical properties of steel 5: plasticity


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Mechanical properties of steel 5: plasticity
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Cooman, Bruno C. de
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The fifth 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.
The Graduate Institute of Ferrous Technology (GIFT)
Game Volumetric flow rate Ramjet Hot working Computer animation Volumetric flow rate Cartridge (firearms) Material Material Last
Ramjet Hot working Sizing Computer animation Cartridge (firearms) Train Book design Material Material Kopfstütze
Air compressor Hovercraft Finger protocol Turning Sizing Computer animation Plane (tool) Plane (tool) Negation Book design Kümpeln Steel Kümpeln Material
Extraction of petroleum Nut (hardware) Stagecoach Kopfstütze Avro Canada CF-105 Arrow Sizing Computer animation Cartridge (firearms) Plane (tool) Spare part Plane (tool) Negation Book design Kümpeln Ship of the line Kümpeln Material Gun
Typesetting Kopfstütze Schubvektorsteuerung Scouting Gedeckter Güterwagen Angle of attack Turning Roll forming Computer animation Cartridge (firearms) Alcohol proof Plane (tool) Plane (tool) Kümpeln Kümpeln
Typesetting Hot working Gas compressor Tau Air compressor Lapping Gas compressor Kette <Zugmittel> Gedeckter Güterwagen Flap (aircraft) Computer animation Cartridge (firearms) Spare part Spare part Plane (tool) Book design Steel Material Kümpeln
Typesetting Hot working Schubvektorsteuerung Printing Finger protocol Gedeckter Güterwagen Computer animation Cartridge (firearms) Spare part Hydrostatische Beanspruchung Engine Ship of the line Cylinder block
Hot working Computer animation Spare part
Hot working Printing Computer animation Cartridge (firearms) Spare part Typesetting Kiln Hydrostatische Beanspruchung Wire Material Steel Kümpeln Material
Hohlzylinder Cord (unit) Plain bearing Angle of attack Game Computer animation Rolling (metalworking) Plane (tool) Steering Steel Material Kümpeln Material Boeing 747
Typesetting Piper J-3 Cub Air compressor Scouting Texturizing Spant Angle of attack Game Ship Roll forming Gemstone Computer animation Cartridge (firearms) Rolling (metalworking) Plane (tool) Rolling (metalworking) Steering Steel Steel Ship of the line Material Sheet metal Model building
it so good morning last lecture we talked about plasticity and how for plasticity we to describe theory to have a theory of questions that we need for elements but an element of that dispersal that describes the elastic stress strain relations so we know we have book lawful that we need a criterion for yielding we need to know at work the stress level that we get plastic permanent deformation we need to have a hardening rule because we need to know how the initial yield conditions will change as we deformed material and we also need a flow world of flow ruled that describes the evolution of the plastic strain during the defamation as that and and and we'll see that that's there this flow rule but in the case of plasticity will connect plastic strain increments to distress all right so far well
the hooks law we know hooks 1st condition we already know that we have with the radical elements to do elasticity brands in the case of Union acts stress if I expressed my books a lot in terms of principle stresses up we find that if I if I express is incremental way the small the amount of the increase in distressed result in small increases in strain Epsilon tens of I can easily computers for you in the actual conditions training conditions and that and using the equations here of some can probably using it for a
long time already and so it when we discussed of plastic deformation we have 2 things to consider 1st let's look at what happens when we deformed this little square here and it becomes this dashed shape you geometrical shapes of this defamation it I can cut it up in the the number of events her so I can have a volume Change separately and then a shaped charge of in this case there is a change in size you can see of this square and here is no change in the side of the square because the the sharing I have the material that used to be on the left side here this wage of material is now on the right so there's no but the changes in the volume of and and in addition of this final shape can be rotated and translated to mimic an actual plastered information right at the let's let's not look at specific there situation where we're asking ourselves with which the year of the work that we need to do to achieve this volume change and the shape change well let's look at the volume change here but strained the material in this direction but it will get longer and ex directory and I will carry out the work that which is given by this well-known equation half of the start from rest that I applied on this side of the few times the strain in the ex-director it is right now do a shape change of this alleges sharing little India experiment in using this shear stress the effects 1 and this is the work that I have to profiling In this case I have clearly volume changes this year he said don't have evolved in the shape or so we're going to need
to say something about the stresses it immaterial Nos and I we we are going to do this by looking at the well-known approach that you probably know from your undergraduate lectures and that is the more Circle approach and that more Circle most of us have seen it uh we've learned how to use it but they're very quickly we've we've we've really lost the ability to use it because it's rather complicated out there in terms of orientation and then you know there's things where you you you if you want to know the stresses you have to rotate into it directions different from the direction of that's on your grass and and then you have to sometimes defied the by 2 and it's very end and so we tend to forget how it works and that we don't use more circle but there is a simple way 2 to use more circle that where you don't have to worry about these effects and so let me explain this how it works I'm so so from now on you can use more Circle whenever you you needed them at what for sold 1 of his why we have problems using more certain because I'm 0 and we need to have make conventions about positive and negative stresses me while I took and the convention we use it as it is before the stress-strain so if we if you look at this little of if you here seemed on its side we read personal defined 2 planes we defined this right playing here and always had collected each place called explained H for horizontal because the uh normal forces are horizontal and this plane here on topical the veep alike as the plane at the vertical because the normal stresses are vertical on display the Kenyans you have the right and it is a little element in 8 of the tensile specimen or or another piece of steel that's being before elastically in conditions of I playing stress we the situation is such that we can't ignore these stresses that are perpendicular to this when the projection here that the screen of as took this number 1 so we define and playing in the plane and then we say that the when the normal stresses are pointing out outward yes so it when we talk about tension then we have normal stresses that are possible if that we talk about compression of the stresses normal stresses compressed this little volume than we said they'd been the signs are negative but it makes sense with the but that the convention is more important for when it comes to the shear stresses when the shear stress on the explained points up really so it's a positive future if it points down we say it's a negative shear stress can uh and uh if it if it points up in this direction and then into the plane and points to the right and these are positive to positive shear stresses like convention from and 10 so if it if it .period downward on the state's playing then we call it and negative the shear stress because the book so let's you do 1 more
thing I have With this convention the elements of convention is when it we are ready to draw more Circle we I haven't axis as barbaric the x axis X axis Excuse me where we were a plot the the normal stresses nose and a Y axis where we will block the shear stresses and it would is important is that the positive direction points down because of its but something you you have to to remember positive directions .period down so what what we basically have in this In in this playing here in this plane stressed there is any of that stress conditions it can be a is is 1 . 2 so if I have to work so I have the my stressed taxes and my share stressed access any I need a point in this craft that defines the stress conditions and did in order to do this I need my more Circle and and it took to build up my more Circle that's 1 thing and the 2nd thing is that I once I have my Moss Circle for specific stressed conditions I can Of course not only have my stress situations in on the faces of the small Q In a material with this orientation but for any other orientation and so I can find out in 1 direction I have only cancel stresses or in 1 direction and have the highest she here stresses get so the best way to do this is to up to 2 give you an example to look at the size
of a 1st that you have a stressed you know stressed great state of so in two-dimensional plane stressed that means you have an X Sigma expects the Sigma Y Y N H share the stressed suspects 1 and the shear stress Y X decide In inside OK than at seduce 3 points I will help you form or withdrawal a Circle and we can do this by computing the dimensions the size of the principal stresses and the maximum shear stress and these are the formulas the principal share you have to principal normal stresses signal 1 1 segment of which a given by the formulas and the maximum share the rest is given by the mean value of these 2 the principal stresses so if I have the condition of stressed state yes this is this point here it has 8 segment expects normal stress the towel ex-wife shares dropped and then I but so built the the distressed state between distressed on the on the plane here yes no yes and that that is here so that this segment why wines and tout X 1 that is is position here and I ended the Moss Circle and is a circle that goes through these 2 points so if I know what signal 1 a but no 1 1 instrument to R and Telmex are I can easily find the center point of the circle and the radius of the circles With the radius of the circle this equal to this maximum shares stressed OK so let's let's see how this works out in practice I have a stressed state sigma expects a speedy major Pascal Cygan were How XY 60 make a pass Council distressed state that measures foreign Minister Little element here he is 18 as escorted and 16 for this plane so 60 is on besides it's positive Was it so that the arrow for part the cigarette X X is it's positive so it's it's pointing normal and to the right on the beach playing on the White Plains I have minus 44 the normal stress it's a compressive stress and then on the negative excuse me the division to shear stresses that the negative that's important the negative are so now and you can calculate segment expects Sigma Sigma 1 wants England to 2 and Max it didn't tell matters here and uh and this is the situation with the values indicate so that once we have our server Sept yes can we find the people of the pulled pork pulled .period is very simply so
distressed state here corresponds to this plane and distressed stayed here corresponds to this display so that but this is that what we call the the .period it's the distressed state on and this situation and this is the age .period that is stressed state on this planet so we take we drop a horizontal through the age .period and vertical to the people that I am and that gives as the people isn't always the people right and now if I want to know the distress trade on any elementary what volume of surface at other orientations I want to know what's up with this stressed on this display was the stresses on display or what is the orientation yes of the Plains where I will have the largest share stresses but where is the orientations where I'll have them as heroes the shares well very simply if I connect the call . 2 this point yesterday of the Sigma 1 1 years I find an orientation where they there are no shear stresses because it this condition of the books Rice this I find baby the normal stress In the extraction of 104 and in the wider erection of minus 46 of these given by 2 red .period and there are no shares dresses 2 and a half if you were to draw the line there In Moss Circle there for these conditions this is you will find that you have to be the stresses are according along the principle that have principal direction I'm in what conditions do I have the maximum a shear stresses in the material there as well here is the maximum shear stress if the I I continue this direction I'm and you can see that means the shares stressed here is the minus 40 C minus so as as we go from but this surface and then we gradually change its orientation yes I go from situation where is by actual stressed by accident I have shear stresses that I have situations where do establishes stress like this then I have the Schuster's becomes very small and there is no shear stress and then the shear stress becomes negative but it is considered on the stage claimed distress negative and indeed but here we are in the negative situation and you see that in this case I do not need to know you to guess this ongoing for divided by 2 I can just read off the stresses the shear stresses and it depends of stresses that in this thing but please note that when you have a maximum shear stresses you still have you can still have a normal stresses however in the when you when you are a
long when your stresses and strains on our along the oriented along the axis of the principal stress but then you have no shares stressed that we have normal stress well I that there was a general
case but now let's let's look at the end of it and this is even simpler in the case where we would have a tensile test repair so in case of a tensile test and the the the stress actors is already along a path principal direction so in this case my little element is here and I apply stressed this in these horizontal directions as positive stressed it's not compression it's it's themselves 1st and have so I had and and there is no stress in the wider action this and because by hate my stresses are oriented parallel to the principal directions I have no shares stresses the Sigma ex-wife single 1 2 is 0 so this is the the value of distressed In the next election but in the white direction it's 0 so I'm here and is no it's the shear stresses to take into account not now I'm going to the rotate this that element and in that come to a situation where they try to find at what angle I reach the maximum share scripts well so that means just tilting the queue here's along the Oh yes 1st before continue the people and in this case is here this is the same as the people has so if I if I know tilt this element was the H playing is here because and I see that the the value of the shares stress increases you can see increasing and then reaches a maximum at this point OK at what is interesting is that the this article is because these 2 lengths are saying this was 45 degrees and indeed that's what we know is that if we have a tensile test the the plane on which the stripper bread the sheer strange stresses accused sheer stresses are maximum are inclined at 45 degrees and you can see here that it works nicely again noted that when when you your that element would use this it is not oriented to give you the maximum shear stresses there still normal stresses acting on these on the faces of Kent right so how however there are things in the wee dimensions so if I have 8 of metal that's that subjected to the number of Francis knows external forces that I knew it can be shown that the at the this distressed state he is can be represented as an ellipse yeah lapse like this and the 3 axis of this lips the form but in Porto going set and which are or which are oriented in which which give me the directions of the principal stresses don't I can't we presented at an end but in case of In the case of the the Moss Circle which which you basically looking at is plane stress conditions and then it as you know that is as you have more circle right and the Moss Circle is in fact nothing else in this section through this speakers stressed the lips but revolution so and I can't begin working with this distressed ellipsoid is not so convenient weaken used an overview of that we can get a look at it in principal stress and we can also use stressed diagram so if we look at principal stressed state so we basically have the in that space we have access which are aligned along the principal directions there's and the stress state that that is distressed on a particularly plane here this is defined by this vector from the origin to and it has components along the x the wide and disease direction we so all alternatively you can think of little planes yes planes and for specific stressed state you will have 8 the component normal to the plane and a component share component in that place support for any stress situation and we In addition to this make it more Circle construction for
3 dimensions in 2 dimensions by having a Moss Circle 4 each set of 2 principle stresses and have had a circle to Circle 4 for instance in this case a dentist signal 1 and segment 3 signal 1 Sigma tune and Sigmund 3 Singer to and signatory nations and as long as my stressed state defined by the Taliban and signal and news is that within this larger of the 3 circles my material will it's really not a norm before yes if it or not to reach each specific maximum shear stress we so that lets them know perhaps have a look at specific conditions so so we know what is meant by these 3 diagram of a distressed ellipsoid the principal stress space and more stress diagram of more stress diagrams is in 2 dimensions but so let's look at the stress ellipsoid if we have a homogeneous tension or compression means that the material is comprised or under tension in 3 directions and stress on the already the pressure is the site and this case are stressed ellipsoid um was so we know the adapted this the principal stresses are equal and so are ellipse becomes the those fears In the stress space years of my the stress conditions as I have negative value the called to -minus speed 4 X 1 book forum along signal 1 Long signal to Anlong symmetries of its point they have basically as that that is the stress conditions in principal stressed and in the more stress diagram years but I should have 3 circles In which are defined by the differences In the principal stresses and what well they're all equal so were on the x axis which is having a single point instead of to the circle but now let's look at the system that's of interest to us as the new natural tension there this union actual tension this is that in reality In the In for the distress ellipsoid would you can think of that is basically as a Y has and half the length of this is equal to the applied tensile stress segment to and signatory of 0 . you have a very very narrow cigar you want with it 0 0 radios because in its principal stress space signal 1 is the applied tensile stress and then I have 0 0 Sigma 2 and 0 signatory that's a point in principle stresses space of course and stressed the diagram lost arrests diagram I have bait watch the this principle stressed equal to security and all into 2 other principal stresses are 0 so on their here at TI origin and the 0 so they should fall on each other OK let's make it slightly more complicated let's make it a buy tensions situations so that means I have now has a little volume element where I applied the stresses in 2 perpendicular directions and so in this case admit 3 is 0 so instead of having a hand in laps I kind of have a very flat the lips nose and depending on but Sosa's signal 1 is equal to sing the 2 it would be a circle Avery flaps because he would in the principal stress space I have now my stress conditions signal 1 along this principle direction and Sigma to along the perpendicular directions to this city this means that stressed and then stress diagram this year we have to be careful when we we use more circles never forgets that uh although signatory is 0 it should be taking into account right so so we have 3 we have a not to principal stress but 3 principles trusted myself to this these are authority Morris circles the
end we can't were not going to go into it but obviously the plane on which we will have the maximum shear stress in this condition is determined by the the this is the circle more Circle defined by signatory distress in symmetry which is 0 along Sumatra's 0 and signal what In the UN goal Peter here years is equal to that it gives me indeed the orientation of the plane where I will see the largest shares stress and so interestingly enough right not influenced by the value of Sigma too foreign good so you know we kind of more comfortable that sigh with stressors and we can define stress ellipsoid underwritten stressed based on where do we you have as axis of we used to principal directions of the stress and we go back to the the problem of shape change and volume change during deformation of of a solid piece of steel so we have we're going to look at the the energy to do for this piece of silicon to look at the height of static part of the this energy which is related to the the volume change and we're going to look at a day for Europe but Torre part of the energy which is related to this change of strategy OK so if I applies the equation we saw earlier 1 have epsilon times Sigma In the energy the needed to change the volume of their small fuel pull it along the x-direction has enough generalize stepped in 3 dimensions began expressing using only tensile using and coordinate axis where we can ignore which allows us to ignore the shear stresses yes I signed for the hi to static for the year for this this energy of deformation of the volume Change 1 have segment extends epsilon next the signal light at long why segments the construction him and by 10 get in an equation for this standard chain which is only dependent on stresses by based by substituting as for the Epsilon's here the the relation given by hopes of so Sigma X Is that epsilon X is replaced by this 1st equation in hopes wall and into the same for excellent white epsilon C and at doing this I find this nice symmetrical equation 1 over To Sigma X squares in white square citizens where minors 1 already question ratio Sigma extant signal why Mama and secrecy and Sigma Sigma widened and they should be this a time segment effects of please correct this if you are using these notes segment should signal X Is that time signals X and I will correct this in the notes that online so we're going to the work further on this formula but before we do this week going to introduce what we called the Theodor stresses and
to do this we 1st start by defining a mean stressed means dress for the average stress is is basically the sum of X Signal accident widened signatory divided by the very reasonable way to define the means and then what we going to do is defined deviate Tareq stresses which is basically the stress In a particular but principal direction minus the deviate Torre and so segment X is that means value times the deviation from this mean value soda and gold is the Didier Torre stressed that x-direction why directed the direction is nothing else and Sigma X minus segment signal y minus agrarian signals he might not bad the has the interesting properties this year this approach will 1st of all let's just assume that we only have the view that the Heat hydrostatic stress that this segment AB is equal to segment signal Wyant Sigma St. Louis and so in this Davies but the the do the work done by this distress you have this kind of stress conditions where there is only a hydrostatic stress that the the energy Inc former B defamation this is given by this equation this is a question and will come back to them no situation we've explained up to now we just had was very general let's see what the viewer torrent stresses are intentions we would have again our simple consultants so the deviant Oryx stressed this Sigma X minus 1 3rd of Texas Sigma experts in a wide receiver where intention signal y and Cigna's E R 0 right so signal that its prime stressed is two-thirds of Sigma X Signal Prime is one-third of Sigma X and Cigna's crime is 1 3rd of certain acts with a minor side To the sum of the Deitrick stresses it's 0 yes and that's of rather general it's here we did the right thing for a tensile stress situation but it's actually the general rule is that there's some of these media or stresses this year and we will use this property in them because so OK so that's a lot of the math here but uh and equations but let's let's have a look in the principal stress based what these deviant Oryx stresses mean no took part in them stress space the stress conditions yes this is a point for instance this point B it is the stress condition it means it's got Texas this is why and as the court in this space and what we do and when we use the deviant Tareq uh approach unitary stress approach as Have the Didier stresses component in the 3 directions he direction in the x-direction in the wider region together they form the hydrostatic component 2 0 stress state and then I component deviant Tory component Sigma X Prize which is Sigma X minus signal average along the x-direction Singer why Prime along the why direction and the signal z along the this is the direction and so this vector here that connects to be that's a deviant stress situation I you see that being is also equal to Sigma X signal wives accusing and so the sum of the hydrostatic and interviewed torrent stresses are the same as the original stressed state and and again you can clearly see that deal notorious is segment experts this distance here that's the sum of the deviant Torre plus Sigma OK so it's again let's in Union actual tension but we only have stressed along the x axis along this this signal on this would be the applied stress so I can decompose this in a stressed that has one-third the component that one-third of Sigma X in the z direction one-third of
Sigma X and X direction and one-third Stan segment exon he in the white erection right so that gives me my hydrostatic the stress components and then the deviant torrents it is this 1 here is has has as components of the Sigma X prime which is this the signal wide crimes in this direction and Cigna's you Prime in this direction you now what is interesting is that this vector that Torah is perpendicular to the hydrostatic component so don't with this illustrates with the unit actual pension situated at any given state stress can be divided in hydrostatic component deviate toward stress .period news 10 right so and what will see is that the hydrostatic component will be responsible volume changes and the deviates horror component is stress component which is responsible for shape changes but 1st let's do a little bit of the mass to show shoulders and uh and also see how this connected to yield but if we calculate the volume change in From this little block here and say this this block has the volume 1 unit volume that it that's a starting forward and we strain it in the x y and z directions us and we get a new volume volume so what is the volume change while the volume changes Starting volume times these strengths and I applaud and uh Sociedad's is not the volume change with the new volume and is 1 plus this was 1 plus the strain of this 1 plus the strain in this direction 1 plus the string and that because segment the epsilon of lines on the ovaries small the I can approximate this the prodded by 1 most this some of that capsule expert to away from the scene and having this by I debilitation which I define as the volume Change divided by the original volume is this summer epsilon expressed to like to and that is non-zero all in the case of the elastic deformation have appointed by and the but it is 0 when we have a plastic deformation but so we are going to look at this the this summer to distillate Taishin and express it in terms of the stresses and together we do this using coaxed wall and we know that epsilon X that's online CIA given by these brief equations x y and z again parallel to principal directions so that the long experts warrantless is equal to 1 over X Sigma experts at the White House and that this all the terms 2 the times that was a racial divide that he and the same so so we know do 1 more step we are going to the EU's make use of the deviant Torex stresses that and so when I do this when I do this I replaced Sigma X by Sigma 8 plus segment X the prime minister I find times segment 3 times signal plus Sigma ex-prime signal y plan students and this is some some of the Deitrick stresses it's 0 so this equation it is very much simplified to this 1 so the volume change and when you do an elastic deformation is equal to yes an elastic deformation of any type as this is entirely due to the mean stressed only there's so the and of course the the value of the elastic constants is 3 times 1 minus 2 times the queso racial divide by the Times segment the the hydrostatic component of distrust and of course but you can see that the this year is nothing but the bulk modulus or 1 over the compressibility but so we basically no what the engine is that we need to have to changed upon you know change the Folio I basically need to I have this equation this here this year multiply
its weight segment this
distress and divide by 2 so this is what I guess that is basically the work needed to do the the volume change it and again I can change make use of the day relations between the segment a B average stress and Sigma X and Y and signals the 2 together this equation for the volume change energy and so this allows us now but because we know what is this the general energy required for defamation you have before yes the Fourier for defamation and and we know the part of the volume Change Of that work so the actual shape change you deem the energy needed for the shape change can now be computer unit that as the son of 2 there's a difference of the the total energy required minors the volume of change energy required but so and if we do this so the last the part here comes from this equation and this part here comes from equation which saying we have arrived earlier which at
the very beginning In which is which is this equation here 1 over to it's 6 to 1 1
earlier if this this equation
here and that's the total energy fall shape change and
volume Change where to now
we're back here this week we need to do a bit of manipulations but at the end you find the and it clearly needed for the shape change is given by this very nice and simple equation if we again do not have to take the shear stresses into account and now we look at 1 of the many criteria for the initiation of a yield of plastic deformation and a things was that of phone users which we heart can actually use in many cases when we're out I'm thinking about the formation of steel it says it's a maximum distortion energy criteria so the maximum shape change energy criteria and it says that yielding will take place when the the amount of distortion and given to the material reaches 18 the maximum value and that value it is the same value it would take to to this story To start yielding immunity actual test so you have a material the same volume of material and I was stressed to let and if I have applied in the work that I do has reached the same level as the work it takes me to let's be cancel yield yes per unit volume and 2 cases of a handsome volume its cost per unit volume so if these 2 rounds of the same then bye bye reach yield so what do we do yeah what do we do well we we we could do this the shape change Energy Inc was so now we're going to say well but let's just put in the conditions for simple Union actual himself says so in this case signal wire cereal and Sigma Siasia said this equation here and and Sigma X is equal to the yield stress so the In the energy needed to have a material yield in you know the actual defamation it is given by 1 plus new divided by 3 epsilon times yield strength square right so and so if I look at these things and this is for the general deformation that 6 the general stressed state and this is where the union actual stressed that I can see that these constants will disappear except for factor 1 one-half and I will get this nice I yield stress yield yields criteria this this this factory here in this segment next month 1 where single women to wear the segments minors Sigma X should be here not yield stress where his people to segment of the please correct this thing you notice should be segment acts if you this may not be the best moment it 2
introduced this fall I
see that we are missing
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this equation here this equation there is nothing but the equation of a cylinder the a cylinder indeed stressed space when access the axis of the cylinder along the diagonal of the Of the 3 access here and it basically tells me there is the stress state there's is inside this cylinder the material will not yield if my stress trees outside this point I will material will heal world classically before him 1 of the interesting points 2 2 look at the that In this stress space is the intersection of bears cylinder announced with the this is 1 we have to give up the intersections 4 situations which we call plane stress situation the yield surface becomes yield function the signal was signal story of a plane which is the analysts so because segment to it's 0 we call this condition plane stress when stress conditions there is no no stresses in the 3rd dimension and yield surface looks like this that is very important yield surface you line if you want and because it corresponds to situations that we often encounter in practice with steals In a very often we're dealing with rather thin shells yes where there is no stress on the Net In in the 3rd direction and 3rd dimension Maine stresses in the plane Of the shelf the there is not much variation of stress in there in the dark in the perpendicular directions to Sigma 1 in signatories no let me show
you that was a case of steels for instance for a number of different steals the this yielding behavior has been studying in detail are and we find indeed that the yielding units the yielding the condition is very close 2 a to this equation here this equation and it tells us basically that this the concept of on To tell us that you understood the material will yield when the new mount both defamation energy has reached its per-unit polymers into straight same level as the amount of information in the actor yielding in the Union actual tensile test that concept basically holds 4 steel most of the time good have now it if we would look in great detail at these yield surfaces but we would see some interesting parameters in 1st of all the stresses along this direction or along this direction years are you in the actual stresses so the these points here on nothing else but do Neil stresses yeah simple tensile situations yields and then this sections the difference segments here yes we presented Of this this diagram we presented different stress states in this quadrant I that segment exon signal why are positive and intention in this quarter of Sigmund X is intention and signal wife compression and so on this a quarter of the year stressed space is easy to the study that's where you have data in this corner because we basically have held by actual tension situation you can I'm usually what we see is that of a again the points the line very well along date the the day I die there and the letters that goes through you how ever if you look into the details very often there are some deviations and will come back to this later lecture but this is what it was type of deviations if I have for instance the sheet metal cold rolled highly formal steel is being tested on and I measured the mechanical properties being in samples that were taken at different models to the rolling direction as shown in this end In the samples I measure a property called the R value and are of value is nothing else but the ratio also the With strain over the thickness strain during a plastic deformation of so so if I have a the tensile specimen here for instance and I pull it in this direction so that the forms they did the thickness strain and the with strength we are not the same very often use and we call this the strained over the thickness frame we call this the aura of health and the fact that they're not the same that property is called normal and I yes no I to it doesn't it doesn't this section is introduced in the same at the same rate and the thickness and in the way so this property yes can however also be sensitive to the direction of in which should take the specimen in the plane of the ship so you also have planar anisotropy and you can measure because and if you make sure that different about this far value at different I'm going to find this quite complex behavior the reason why we have this complex behavior that's due to capturing and the texture in itself is the result of the processing of review cold-rolled steel and you do we crystallization annealing afterward she noticed that the aura of value will increase which amount of the reduction that you give and will also did the way in which the material is an isotropic in the plane will also change with the amount of information this hazard big impact on the are yield functions because it means that when materials are strongly isotropic as a consequence of words in this case Crystal graphic texture that has an impact on their yielding behavior and we I have to use adapted yield criteria we can but we will talk about this next time we meet thank you very much for your attention


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