Mechanical properties of steel 4: stressstrain relations
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Formal Metadata
Title 
Mechanical properties of steel 4: stressstrain relations

Title of Series  
Part Number 
4

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 fourth 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 stressstrain relationships including constitutive functions, yield criteria, plastic instablities and strain rate dependence.

Keywords 
The Graduate Institute of Ferrous Technology (GIFT)

00:00
Photocopier
Nut (hardware)
Ballpoint pen
Club (weapon)
File (tool)
Tin can
Machine
Scouting
Buckelschweißen
Volkswagen Golf
Mountain
Computer animation
Screw
Ship class
Doorbell
Remotely operated underwater vehicle
Steel
Engine
Material
Engine displacement
Model building
06:38
Sonic boom
Volkswagen Golf
Computer animation
Cartridge (firearms)
Tau
Photographic processing
Remotely operated underwater vehicle
Gas turbine
Steel
Engine
Ship of the line
Material
11:08
Volkswagen Golf
Computer animation
Spring (device)
Steel
Material
Model building
Model building
12:26
Stitching awl
Airbus A300
Cylinder head
Volkswagen Golf
Alcohol proof
BMW 501
Spare part
Basket
Steel
Engine
Internal combustion engine
Material
Model building
Mechanic
Ampoule
Train
Scouting
Outsourcing
Computer animation
Cartridge (firearms)
Fahrgeschwindigkeit
Work hardening
Gas turbine
Remotely operated underwater vehicle
Steel
Model building
23:41
Nut (hardware)
Lugger
Merinowolle
Airliner
Volkswagen Golf
Stardust (novel)
Tire balance
Spring (device)
Institut für Raumfahrtsysteme
Spare part
Tool
Engine
Material
Diamond
Ship breaking
Tin can
Fiat 500 (2007)
Ballpoint pen
Train
Machine
Computer animation
Brake
Fahrgeschwindigkeit
Cartridge (firearms)
Thrust reversal
Remotely operated underwater vehicle
Steel
Diamond
37:19
Mechanical engineering
Hot working
Game
Volumetric flow rate
Ship class
Doorbell
Typesetting
Spare part
Endeavour (yacht)
Book cover
Material
Surfing
Ford Focus
Mechanic
Chemical substance
Computer animation
Work hardening
Photographic processing
Welding
Plane (tool)
Remotely operated underwater vehicle
Gas turbine
Material
Steel
Diamond
50:08
Typesetting
Hot working
Cut (gems)
Tiefgang
Air compressor
Train
Finger protocol
Circlip
Ship
Roll forming
Volkswagen Golf
Computer animation
Plane (tool)
Plane (tool)
Negation
Gas turbine
Book design
Remotely operated underwater vehicle
Kümpeln
Material
Kümpeln
58:15
Mechanic
Scissors
Weapon
Ammunition
Belt (mechanical)
Volkswagen Golf
Computer animation
Plane (tool)
Plane (tool)
Mining
Kümpeln
Ship of the line
Kümpeln
Material
1:02:43
Saw
Computer animation
00:01
a without were so mean just a few points
00:09
before we go along on the
00:11
material and In a
00:16
related to last less so
00:22
how to apply these equations here who might be pleased if you want if you learn from printed copies of what's on the on the class nuclides files you may want to check the the 2nd file updated yesterday afternoon and some extra material on some of these fight in the club what I will show you a now not essential but it's just nice to have you felt you only be hearing called the kind we for instance so when you are you know when you might attract stressstrain curves status the B the machine doesn't measure stressstrain curves right the machine does something with what it matters and distressed right there should not forget that with the machine actually does the say we have a tensile machine the screw driven tensile machine now and you have a business of regular sample here In script that you would His latest partners and moved yes and then you have the screws system here which is driven by a motor yes and West Coast which moves up at a constant speed right that moves at a constant speed a controlled speed now and so on and then there is a load cell either here or there that measures the force and so what they and and then you as a researcher at the maybe you put a strain gage here now that so so the machine measures probably resistance and the force you write little voltage as a function of time and that gets translated into forests and displacement yes and then the calculations start no displacement is translated into strain forces translated into Strasse yes and so on and it's all you know certainly if you're involved in research it's always interesting to look at the origin of data the Commission also gives you are nowadays to the forest the displacement as a function of time because there may be some interesting In revealing details there because when you seek stressstrain curve it's no more time funniest restraint of time has disappeared however there was the time was there there is time hidden In the data so if you are you can see funny things in your stressstrain diagrams always worth looking at what the actual data was mainly 80 the at times the affirmed that the stress or the use of force as a function of time and strain or displacement as a function of time very revealing about going away so far so the the uh if you can calculate also yourself based on the of the engineering stresses and strains yes that of which is true strain areas and which true stressed ourselves and and and just as an example here for instance you go through the data you find a maximum forests provided difficult to see because some some steals extremely flat can I areas India where with the forces macros houses are found to 272 I put it in this the 4 minutes yes and at that of their right to 170 to 1 plus he of the engineering strength there and find 340 mega Pascal's yes and so on I also see there indeed the uniform the longer patients but the corresponding troop strength at the maximum is not 25 and 22 per cent to in general on you see a reduction in the uniform along the nation's engine for illegal from engineering to true strain and you you get an increase in distress when you go from there engineering to true stress the but so again very important some guy comes up to you and says 2 thousand make a passed ball what is that through stress for engineering stressed that's true stressed but this year an engineer would say I have 272 the the professor would say I have 340 yes the 340 doesn't count OK is but in an engineering situations you it's the UTS accounts are but so again very careful with mechanical properties of which ones are you reporting but it was another thing that's
06:41
kind of usually skip but that's Richard you remember the there is the instability criteria right just said you know if you knew defaulted in materials development that's the 1 you have to look at right because that's the 1 that really tells you fundamental things right again not saying that the engineering things not important that they are important in their own right but if you if you're materials research the the socalled goal see their criteria is is the most important that there is something initiated many textbooks is that OK and this is of course again construction which is rather odd things but anyway I'm just in case you hadn't heard about it and then you were on looking into it so if if you will if you take the the this stress To the strain at maximum you can you can we write this interplay of our through stressed the right to the engineering strength tests you just have to then multiply with derivative of the engineering straining to be true I stressed that this didn't you you wouldn't what you would would be Epsilon and you find is rather interesting the result that the derivative of the this be engineering the true stressed to the engineering strain times 1 plus the engineering strain is equal to this distress at maximum when the forces maximum I personally lover use that but it's it's it's a nice thing to consider because it's it means that the brevity of the truce stress with respect to the Engineering's terrain has found that the the derivative signaling the not segment maps along with me that is equal to the distressed the mathematics but the true stressed that maximum force write the maximum of the I engineering stressstrain curve gets divided by 1 must be so that means if you if you think the the current calculated that you take the relevant for the point corresponding to the maximum in the engineering stressstrain curve that then on this line here is 1 of the less he obviously this is the this is 1 and does yeah and that's the get criteria but that sponsored the construction and I like I really want to get into good save the ones because some people confuse it with the concede there criterion it set yes so I this has nothing to do with the conceded criteria this because constant for don't try to and no 1 will try to get it use this equation to to find the derivatives to the true stressed we think this is a derivative to it and another curve the truce strain which was stress expressed as a function of the engineering stressed that whatever I do I personally never use that I thought I made this point just in case and that the new 1 had ever gotten confused or this was going to get confused about this this construction and I've also never seen used actually this it's in textbooks and themselves 1 of these things but so so we did see all these conventional
11:09
along and I denied it reminded you of 2 the facts important that whenever somebody reports you'll strings and tensile strength you know you also need to know to what kind of conventional stressstrain model it was
11:31
fitted in front of them that's the 1 that most often used this thing that you can assume when people talk about you'll spring Pennsylvania is that this too soft equation in which which looks like this and and that usually gives you 4 steals of Riddick steals and values at around .period To something right and and end to the kids if you determine these these parameters you can easily for instance determine what you'll strength tensile strength uniform locations are friends but I also told you that people have developed empirical equations which allow you to calculate these parameters as a function of a composition and some structural parameters and end up in
12:30
as it is the 1st of gas for but still design it's very good news you can use the stick with it as I said last nite they're based on a really very large the amounts of data we also talked before closing at last week about the fact that the strain rate has an impact on the stressstrain could depend on and another question is whether worth the impact the clearance In the 1st of all if we use the whole among equation usually In conventional tests the way we take care of the strain rate factor you have epsilon docked To the power and and is not against the material constant it's very sensitive in particular to the strain rate at which you measure it and if you do very small strain rates the strain rate sensitivity is very large so this is an example here at 800 deg C. This is not a room temperature a measurement but it basically means that we can get the string rate sensitivities in steals the box . 5 and sometimes high at high temperature in the steel sector can go up to . 7 I will be talking about this later but the these highvalue sees a high values very high values of string ratesensitive physical and that makes a material supra plaster so to it's steel can also have super plastic being behavior at the end at higher temperatures and if we use off strain a strange way to and again as I said will be talking about this later because the United said just just a moment ago it's at high temperature so it's a little it's related to creep deformation of so but but but anyway tho again and again you know there's lots of knowledge about hightemperature behavior of of steals white why would people be interested in in having this kind of data and why would they have a lot of data as a consequence while because steel is a process at high temperatures we know when you when you make a steel product whether it's a quarrel or bar this always been a hightemperature defamation process has to do what we call Ross rolling or off affordable forging yes and that's always at high temperature and that's where we're at an Austin Addicks steals yes there's a lot of data lot lots of interest in this kind of data because the forces you will need to apply to deform the material Will be function of the strength of the material right so so so people have instead of doing lots of very just develop empirical equations which which give us for instance are perimeter models for carbon steel at high temperatures In the Austin Texas state and that you and values you get and values and then you get a constant for the whole among equation in this particular case as a function of carbon content and temperature in this case right and you can use these the equations to determine the the
16:34
stressstrain curve Of the princess Austinite at 800 degrees on acidic steel in this case of political field at 800 degrees and this is what you would get out interesting a look at this strength values less than 100 make a pass less than 100 remember don't material and asked you to bend his there with about 100 megabytes count that prevented so that at high temperatures and our steals are much softer much much softer than at room temperature there and was that's the reason why we we we do half hightemperature defamation is because I know material itself today will be part so you have to samples yesterday the was 1 of very hard as I told you that's about 1500 make a basket when you that material was made the formed at high temperature it was as soft as the other 1 the will that will go into details why that is and the physics behind us as we go yeah right now it's a little bit more at high temperatures a little bit more complex because there's been a low temperatures because there there can be lots of micro structural changes In the material I as he as strained a material the material will tend to Rick in certain cases or they will be very pronounced recovery yes and that that and as a consequence you you don't get stressstrain curves that quite look like this model calculation here you cannot use this up to very high strains because of very high strains which you see for instance were also denied that the strength peaks and then drops yes and that's the the reason is because you get dynamic recovery to the need to get new new grains that are fully defect free or you get a reorganization of the the fact of dislocation and Mike rastructure and I'm sorry that the recovery and all you get the dynamic with delight this is actual data the source of this mythical these equations Syria can only use them for small amounts of stress Brett said now and I told you that lets us see eye to this is something that we discussed last time where I said that the people see their criteria for instabilities of bit segment the epsilon is equal to its the stress that the maximum forests and maximum load that and so we will weaken if we have a stressstrain the relation like that the whole among equation I just showed you guess which include strain rate pendants and strain hardening constants yes I can now this derivation using the Holleman among equation then this simply too simple matter here where this is being done so the the Sigmund Max is simply Holleman equation where I've put in uniform strain for the strength of then using this relation but between the true strain and the engineering training I can find that there is a relation between the instrument In true strain and the increment in engineering strain has and if we use now the fact that In a mechanical test need but this parameter here if you good to be true the equator where this parameter here the change in the rate of the engineering stressed the strain Excuse me during its test at constant strain rate is 0 the content that is because we have a constant across velocity this factor 0 and and so this allows me to determine the fact that is interesting in relation that derivative of the strain rate to the strain is negative of the strain and so if we go on here we plugged as this this finding introduced the original equation we had here now we find that the uniform elongating an divide by 1 cent and so on obviously is that the derivation of you can go through it but this is a little bit more complex than the derivation you may have seen as an undergraduate but it's basically the same because it tells me but if you have a whole among equations 4 Europe has relation between stress and strain you uniform illumination is equal to end yes and on is excellent and divide by 1 plus ATM yes if the if you take the the strain rate sensitivity into cash so am it's interesting resultant interesting result is that the the energy value yes it is important to increase uniform along variation the end value not so much right because if M is large yes it'll help it has a negative impact and this is a positive and the value now I is is that a big impact while in general not the most deals have relatively relatively low values for being so it's not a big big impact we'll see some numbers in the moment it's not a big impact but strain rate sensitivity is important and I'll show you why OK this
23:43
makes samples why strain rate sensitivity it is important I have here to materials it it would still insisted that the steel which to Bostic steel which has wired strain the hardening behavior large and value and this is an NIF steals the attic steel it's got a much lower and value so as a consequence fears I see that the UTS the place where I have the maximum force yes maximum force maximum load the maximum the engineering Strasse this year at about 25 per cent with the idea steel and at about close to 60 per cent for this yes however there is a big difference after reaching the 25 per cent strain they are still continues to the fore yes In the next In the next for another almost extra 25 per cent it's whereas the other guy up there breaks brakes suddenly and the reason for years a difference in strain rate sensitivity the strain rate sensitivity a lot of the which still is close to 0 the spring rate sensitivity of the IRS steel it is 0 . 0 50 it's not a big difference in but the impact is this actually large the reason is of course that once you get up once you get navigate remember once you get necking all the material that's outside of the neck continues to deform elastically says stops the forming plastic the only plastic deformation is here as a consequence the strain rate goes up I very much the strain rate the machine the velocity of Cross said remains the same year program that it's not it's not gonna change however the strain rate here increases a lot Of course if your strain rate sensitivity 0 0 yes you have epsilon . 2 2 0 yes your material doesn't strengthen yes it doesn't so are you may connect and it goes quickly so that distress increases and you just break the material however if you're and value is positive yes when the strain rate increases the material strength yes 2 you can delayed structure that way this is the how do we know how the hell can you measure this amount and value well you can do 2 things you can go you just measures stressstrain curves at different strain rates the prince's minus 1 per 2nd 10 to the minus 2 per 2nd liners with the 2nd minus4 for minus5 Pesek in what is an interesting 1 this 1 as I lowered a strain rate it gets stronger so this particular material has a negative strain ratesensitive yes as I increased the strain rate it gets softer so again and under the UN I expect that the behavior because you will usually think of when you before material faster since the material gets stronger and annexing Nunavut shown you an example that's no search steals that we used in automotive applications we want them to get stronger when the strain rates go up but you have materials that you just the reverse yes 2 years of course if you do this kind of measurements you need lots of samples right 1 for every strained right and if you do research that can know that can be costly In terms of you may not even have that that much material so if you do research on new materials also won an alternative is to do what which call the strain rate jump matter so you basically do we measure 1 rose stressstrain curve at regular intervals you change b this train wreck the Franzen's here of we were measuring this is justice segment of the stressstrain curves segments of a little that of the stressstrain is here and suddenly he was changed to strain range from 10 to the minus 3 to 10 to the minus 4 would have basically tell the machine is continue the defamation but now uses a slower speed yes for the process and then and then what would we see is that that there is a trenchant yes this trenchant comes from the fact that put the material inside the material the material has to change what it was doing but it can't do this suddenly as suddenly as you change the defamation condition to get a trenchant the situation where the material adapts itself and so you'll usually ignored that part you don't need to ignore that and you win when you measure to strengthen its sensitivity you you will ignore it so you get this this that 1st and then it increases in this particular case and and and so we measure another the stress you increase and stressed that said the strain rate sensitivity is the change of natural logarithm of distress Over the change in the natural
30:31
logarithm of the strain rate which you can and you can calculate simply by With with this formula here where were similar to this I knew it was stressed but after the change industry and similar 1 was distressed before the change industry divided by differences in natural logarithms of his strained right after and before it so that allows you to get on a lot of data points yes and so you can you can get the string insensitive to the material using 1 sample basically no the point about the strain rates instantly brings up an interesting problem certainly in research for and of not only in research is the fact that I I'm always assuming when I'm talking that I'll have him you know I have material that I can test this material I want to know the properties of citizens taken an eye test not very often in engineering or in research you don't have that possibility yes because you know for instance you working on the new material and the amount of new material there is is this bear right when you need it you can't make the stressstrain curve for that material but you may want to know what's in what's the yield strength which the tensile stress in other situations so you have the vast amount of material yes but if you would take a sample you would damage you would damage the material will give you an example for instance In enrolling as you need Brussels the rolling the around in enrolling when you to form a material you need you need big you you have to columns of materials huge amounts of matter and you you really want to know what I would say the property of this material but measuring the properties you would destroy right have to take a sample so but they do Is measured hardness by having metal as a small metal tool yes bounce balance of the finished product that doesn't damage this is the 1st doesn't just bounces off these on the surface yes and then you can correlate the height at which it but we bounce with the hardness of material and then you can correlate its hardness Whitaker will strengthen the tensile strength of them right there are more cases than you think where it making the South specimen is this this is not an option you have in research L we have a pretty good alternatives if you have very little material and and and that's using a hardness measurements factual hardness measurements and the the about this year the best and the hardest measurement to make in terms of research purposes is Vickers yes and what will go into it but it did was down to the fact that you can make you can correlate Vickers tests at 1 of whom very many different dimensions yes and compared the data that with the but I don't so let's let's look at the figures test you would be making a lot of balls a few things about that so probably most of you are familiar with this test but basic comes down to you have the diamonds tool against small ones when they which has a young son so the the uncle of the diamond tip is 100 6 degrees in and you basically press it into the surface of U.S. Steel for a specific amount of time that been using a specific forced yes L In the new you withdraw all the details and you look at the impression when new measure the surface of that inquiry yes by measuring the diagonals that day at the diagonals plus the fact that you know the shape of the the permit tells you the forests nuclides divided by this surface back to value is the the Vickers hardness and now the Reyes a son of a lot of confusion and again you have to to be critical on when when people report data or when you use harder status it's in kilogram for sperm millimeters square right Norman made a Pascal the data the nice thing about this measurement is that there is a very nice correlation between meals staring and tensile strength at the little choice here but is similar to the correlation between the offspring themselves that's pretty Lanier and that you can use to correlate a small Vickers hardness test to what you would like to know have gone is a tense Sultan and this correlation for Steelers win step by many people for many years and and even has the advantage that and it goes from for erratic structures To mark very hard marked acidic structure it also holds pretty much independently of the new on the details of the MicroStrategy that's a very good news and of course if you have a nice but the
37:19
experimental relations like this you can develop equations equations that relates Jordan yield strength and your tensile strength to whatever you measure yes but I like to use I personally like to use this equation is acquired recently published and by the cover very large variety of steels measured over many years now and and they give pretty good results I that so it's a very important again and that also people are extremely sloppy With hardness measurements I don't just tell you partners Vickers hardness measurements and if you value you always should give when you report the the heart of America's hottest value the forest that is used yes bank kilogram force yes not Megan you can make a profit out me Newton's yes any also should give the dwell time 110 10 seconds so 250 h of the yes old .period whose slashed debt that's the that's the correct way of reporting back and so we know it's a it's a Vickers fast that did the the data is being kilogram forced the square millimeters you've used 8 . 2 kilogram force load for 10 seconds because while it's a technology right so you if instead of the new don't tell your colleagues this would be a Brunell measurement or any of the Rockwell measurements yes and no but if you don't know but you know you you'd be up not knowing what to do so anyway so this is an example here for 215 you applied its function is these to equations and you find 628 mega Pascal with the yield strength and about 800 megabytes of offensive transceiver useful thing to have and of course it's empirical rights empirical equation there is in the literature of yes an equation which is very often the use been misused against and that is and it's usually reported that signal estates divided that's how you see it in you have very nice publications and they tell you the segment is age divided by 3 and and I said now and and people you said and done and sometimes they don't use it correctly the Frizzell this equation does not give you it's not an equation for the yield strength or the tensile strength it's an equation the flow stressed the most between 10 and 80 per cent chance of defamation and then he ain't here yes yes style 0 value In May posture so With the beach value is evaluating image of so so it will so it will tell you for for instance the city has multiplied by 10 about that if you don't know that you take this to divide by 3 yes but it's much much smaller than 600 right so very very young and very very it's not convenient you the the debate unless you know it's there and it's a formal I use a very often and and it can lead to to some confusion
42:02
can arise so now that we've we've wound said a few things about plasticity already which said that there exists no conventional stressstrain curves and endeavor but we didn't really look into plasticity that way the mechanical lead inclined person would issues and I so we need to say a few things about this uh about the mechanics and Stickler mechanics of plane stress and plane strain yes some of it will be a repeat of the repeal petition of something you already know but I it's important to put things in perspective of some of these elements will be using this and if we don't discuss them beforehand may be very difficult to do this at the latest thing so when we when we think about plasticity from from a putt from the no fundamental make chemical point of view and not enough to now and I have said nothing about Mike restricted by the way you know I talk about your strength you know I I would ask you Cahill calculate meal strength of a certain steel with a certain Microsoft it's still going it's a wall I all and give giving me to give it to me I I don't know how to calculate but but so so and again the when we discuss the plasticity we will also not going to of details of Microsoft Truchas and composition and things like this but it's it's very important to know that it's part of a big group of things you need to know to go into a mechanical properties of any material so when you describe plasticity From a mechanical but if you need for things for things must be present in other ways you you cannot describe the city took a bite this cup been formed in certain ways yes In if I want to study the formation of this as a mechanical engineer I would use finite elements of course right because it's very useful but I couldn't start doing this yes if I didn't have this theory of plasticity but this serious the surf but the city is built on 4 elements and and they'll look very acceptable to you personally need each need to be able to describe the strains as a function of stress or stress as a version of strain and those are given by Hoechst lost that we know these things there's already then we need Out of the description or criterion for yielding knows that we need to have a point nowhere where they the yielding the elastic yielding walls the elastic deformation will stop and will go into classes at the that's Golden yield criteria and and then we need a work hardening world has we need to know how the initial yield conditions will will change during deformation if you take a material the biggest and for instance test specimen you you while you pull it it will yield at a certain strength against and you come back the day after you take the same attended the material you just acid yesterday you that it's yield stress will be higher will be different and so on so so we need what's called a work hardening rule which is the tale of how does the yield conditions the initial conditions change during the formation and then we also need a what's called a rule which describes the evolution of the plastic straight years and we'll see some interesting examples an amazing tells us that as as you increase distrust how you know what happens to to distract him here and in particular because there is no stressstrain equivalent In plasticity to stressstrain equivalent to a hoax law in elasticity we're we are using increments of strain so With the floral tells me how this strain increases in a particular stressed state nearly all very clear when when we come to this for instance the elastic strains became very simple but this is my my my hopes lie assumed that x y and z are 2 socalled principal directions so I want again and for reasons that I already explained we kind of like to use simple formulas and so we we say let's focus on June the actual tensile test so you only have X Sigma X that's nonzero Sigma why consumers use Europe so I get this behavior for the the increments of this strange as a function of the increments of distress and I can I can write as a derivative of this tells me for instance that in the X direction the increment of the day the the strain goes as increment distressed divided by now that's basically the guns but but but let's now look at just complex deformations solidified look at a little Peace Square material here and I deformity into this Dr. shape well I can get this started shape in this position bye 1st increasing the the volume of this square it's not that the move the anything and I can also after I've changed a volume I can Shearer yes share it was and then I can I can turn it around and I can push it over here is negative chips so I In general the general plastic deformation will consist of volume Change shape change and then rotations and translations yes then now let's say something about volume change 1st well 1 of 2 the bigger loss of plasticity is there is no Volume 2 yes but volunteers a possible in elastic from writes that we already know something here yes that will help us understand the yield criteria volume change shape volume Change the last the fastest time is of the domain of the justices should change the fastest definitely 1 of the things that people do In
50:09
plasticity they look at work the work needed to the form something this the ransoms if I look at this at a clip you hear that and I stranded and xdirection the stranded in the exdirector I've changed a volume right volume is larger now so we can calculate From the books long that the work that I've done the stress finds strange divided by 2 that gives me it's just wonderful unfamiliar with this if I have a stressstrain curve yes if I did this train but this draft this integral here segment at the time From the X from 0 to epsilon acts as this surface that's the work per unit volume of defamation of elastic different yet so it's very simple of how how do I integrate this into 3 seconds while I say well it's this rectangle surfaced signal next time epsilon next divided by 2 right I don't eat much integration right so that's that's basically view but had no let's look at the St. you know I'm just cheering and I'm sharing it volume on the floor of this new volume change because of the volume that's kind of disappeared on this side of this cut it off and I stuck it here right but there's no volume change at all right who pointed up and then just based on the same idea except that you replace this by sheer and despite the display Shearer the stress and this book is sheer I'm going there you get for the good work done to to change to shave you get that what have come to stay the ship where of course dollars a share .period normal forces give me volume changes share forces give me defamation shape changes that's in a nutshell the message that but before we go on like that again just to make sure that everybody is with me I need to talk a little bit about stresses have been solids in when you stress a material found there is it something that allows you to know think about stresses that be normal stresses hit the shear stresses In a material yes and that's called the Morris Circle construction and and derivative and the equations that go with them so said if I have a way of the little piece of material that's subjected to some kind of stress I can always cut out a little bit of that little piece of it it's it's easier said than done and measure distress yes actually we do in practice that's not what happens in practice when you measure stresses you actually don't measure stress you major strains by some methods has and then you derive destroyed the local stresses and I'll give you an example again in that's simple animal but I wanted to share with you an easy way to use more circles because I've always disliked more circles construction because I could never remember in 1 direction you have to turn things to find distressed right at top and because this the person who teaches you Morrison Circle has never used it himself and you know and you know and this is supposed to teach but I haven't tried not my tree but I like to to share with you about how to avoid worrying about Moss Circle and actually the only if you use it if you need to use its use after circuits so that's a little element in your solid I and uh distressed state is decomposed into shear stresses and normal stresses on the basis of this small you know there is a lot more circles construction for 3 dimensions that stress states in 3 dimensions and will be candlelit talking about this but not much but it's not necessary and ended so what would we be talking about is a plane stress situation so we ignore b stresses and shear stresses normal stress in this direction but In fact so we have shear stresses and normal but it so we need to do to things 1st of all we need to talk to define what we what we call H playing and the place that there is a very simple he explains this gap is that this is the guy on the right yes and if the plan is this is the top place you know h plainly colored explained because this normal stress is horizontal and the goal is to be blamed because this normal stress is vertical no 1 of the things we need to do is have a convention yes yeah what is positive and what is negative and that's 1 that's the the 1st thing that makes the Circle very confused construction very confusing is that nobody ever talks about the conventions yes and the conventioneers when we have tension on the surface we detentions of positive news like this of the tension the this stresses are outward we say it's positive if if compressive type we send negative that someone can be negative the other 1 can be positive that's but the direction is important if it's inward it's negative outward 2nd for the shares that's also important what is positive would we call positive or negative the sheer on this plane on the explained is indeed upset a direction we call it possible if it's in the down direction we say negative broker and once you have 1 of the shears yes defined 1 of the other shears are all all the 5 right because the elements needs to be instant stable right of so this one's up this one's down yes and they point to toward to each other right on the on the plane and the explained 1 they point toward the cup but so next step I so 1 of the things
58:23
you need to do it is to make it work is the shear stresses the access has to put good down that's 1 of the important things as they the Act was the positive acts points that a positive direction of the shear stresses that positive at the direction of the principal stresses this is horizontal to the right so when what is the situation now we're saying we know on the on the plane yes what are what the shear stresses and the tensile stresses on are yes we do we measure somehow this then once we know that with the construction of more circle we can determine the stress situation of in any orientation that's basically what it and as an undergraduate and I remember I said This is wonderful no problem and and you recognize I know how this is being done right from the undergraduate the mechanics courses and so on so society you have a stress situation now so so you have measured this dress these 2 normal stresses and this year's stressed on a set at a certain location of new material content and you can construct you'll Laura Moss Circle bye so this is a signal 6 it expects in this direction and the towels to the segment X axis and Sigma and go on from this direction and and this goes in that so you define them 2 to get a circle you calculated the Sigma 1 1 signal 2 2 these are the lines here these extreme .period using this formula and you determined team Max that's this point here that's the difference between these 2 divided by 2 and then you're familiar with with this idea so let's let's see how this is how we used this because so for instance you have a stress conditions x x why y In Tao exwife said this was 14 so this point is was 40 this is minus 20 this point is minus 1 and towel exwife is 30 that means that this yes distressed state for it is is this Is this point on circle and and in this sport and so the circle that goes through these 2 points allows me To determine distressed state all all the planes on all the Q suffused with different orientation and in order to to do this the 1st thing I need to know it's belts the more Circle Nelson and build a more server by determining X 1 1 X 2 to those are the principal stresses and this year's crop but I think I'm over time and this take more than 1 minute so instead of rushing and I'm just going to work postpone until Tuesday we thank you for your attention and your
1:02:40
patience because I went over time yeah as usual