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## Class 11 Physics (India)

### Unit 13: Lesson 1

Stress, strain, and modulus of elasticity# Stress vs strain curve

What happens if you stretch something beyond its elastic limits? In this video, we will explore the regions beyond the elastic limits. We will take a steel rod and keep stressing it until it breaks. We will draw a graph of stress vs strain and explore all the different regions of it. Created by Mahesh Shenoy.

## Want to join the conversation?

- here we can see that hooke's law has limitations..is there any alternate law which overcomes hooke's law? like bohrs atomic model overcomed rutherfords model.(6 votes)
- Among stress and strain which quantity is independent and dependent??(3 votes)
- stress is always dependent on how the system (or body) is constrained (resisted from accelerating). strain is however independent of this.. It is possible to have strains without stress too..(3 votes)

- What happens if we graphed the steel bar's stress for the neck as well? Would it instead shoot straight up?(3 votes)
- probably. Until the fracture point i think it would still keep increasing(3 votes)

- So point A is called proportional limit point, is the point above any applied stress doesn't obey linear stress-strain behavior (Hooke's Law). And the elastic limit or yield point is the point where a combination of elastic and plastic behavior is still there (although it's not in the plastic region) so according to ASKLAND BOOK at that point the departure of the plastic behavior is noticeable unlike Proportional limit where the plastic behavior is on a microscopic level.

So know my question is: why he said if the stress that is applied on the specimen is between both proportional limit point and yield point will return fully as if it's elastic ?! I think that is wrong. Since there is also plastic behavior (small) beside the elastic the specimen will retain but some deformation will be there isn't it ?(2 votes)- No real body is perfectly elastic.

Even within the proportional limit, there will be microfractures (minute).

So, there will be some degree of deformation if a body is strained.

It's just that within B, the material can return to its original dimensions without undergoing a noticeable plastic (permanent) deformation.

Think about a rubber band, it loses elasticity after being used frequently even if strained below the strain corresponding to the yield point.(1 vote)

- why if the strain increasing and the stress is decreasing?(2 votes)
- rewatch the video, he explains it.

timestamp:6:59

if you don't get it let us know(1 vote)

- Hey! Why is the yield point beyond proportionality limit?(1 vote)
- steel still retains its original configuration when stressed beyond the proportionality limit. this happens till yielding sets in (yield point). however the stress beyond proportionality limit varies non linearly vs strain. This gap is usually very small.(2 votes)

- Let's say I have 2 materials,

ultimate tensile strength of A is greater than that of B,

but, the region of plastic deformation of A is lesser than that of B.

Isn't A stronger than B?

P.S: my teacher said B is stronger cuz the area under the graph is larger. How does that make sense?

i.e., if A was diamond and B steel (as in the video), diamond is definitely stronger.(1 vote) - How do I calculate the material thickness needed for a given metal, in

the cylindrical part of the internal combustion chamber?(1 vote)

## Video transcript

Hookes law tells us that the stress is proportional to strain but you can only use this within the elastic limit so the natural question now is what happens when we go beyond the elastic limit I mean we know the proportionality won't work anymore but what happens to the material and that's what we're going to explore in this video and so before we do that if you require some refresher on Hookes law and stress and strain then we've talked a lot about this in previous videos so it would be great idea to go up and watch those videos first and then come back over here all right so we're gonna explore the regions beyond the elastic limit and the best way to do that is by drawing a graph so over the y-axis we're gonna plot the stress and over the x-axis we're gonna plot the strain and the shape of this graph really depends on two things it depends on which material we're dealing with it also depends on what kind of stress that we're dealing with we've seen that there are a couple of ways you can stress the material you can do tens i'll or you can do compressive there are other ways as well so for the sake of this video we're gonna we're gonna focus on steel that's the material we'll be focusing on and the kind of stress we'll be talking about is tensile stress which means you can imagine that we're pulling on this so to create tension and the experiment that we should have in our head is someone like this so we'll first pour and as as the material elongates there will be a restoring force generated and if you calculate that restoring force divided by the area the cross section area that will be the stress but but when the material comes to equilibrium I mean it will eventually come to equilibrium right when it does the pool will be equal to the restoring force right and so to calculate the stress we can just take the pouring force and divide by the area and then we will calculate the elongation divided by the original length W of the strain and once we do that we can plot after that will increase our pool the material will strain more and we'll repeat the experiment over and over again and we would have plotted this entire graph all right so if you do that and today we have machines to do that I mean we're wondering how do you do that so we have machines to do that for us and so if we do all of this then the graph that comes out for steel would look somewhat like this now this graph looks a little bit scary but don't worry we're gonna explore this step-by-step the first thing is how to look at this graph whenever you see this graph and we are at any particular point what you have to understand is this will be the amount of strain generated and this will be the amount of stress so for example whose if I say okay let's concentrate on this point we are over here it means this much is the strain and this is the stress all right okay now the first thing you can note is that there is this straight line there is a linear region and this is the region where Hookes law is working because it hooks log and Hookes law works sigma stress is proportional to strain and proportionality means straight line all right but the straight line ends after some point over here so we could mark that point you could look at that region let me call that point as point a and we could say now beyond point a no longer it's a straight line which means Hookes law no longer works so point a marks the Hookes law limit so we are over here now on the graph so what do you think will happen if we go beyond this point if we increase the strain even more you might think okay now we will have permanent deformation right it turns out no it turns out that even beyond this point a we are still within the elastic regions turns out that there is another point somewhere over here point B and that's where the elastic limit comes in so point B is the elastic limit elastic limit it's also sometimes called as they yield the point yield point so between a and B Hookes law no longer works but we are still within elastic regions that means if we are to if you were to strain up till this point over here and if you were to let go of that d-forming force the material will snap back but once you go beyond the elastic limit I mean imagine you strain it so much let's say you strain it so much and and I should have drawn this a little bit longer but anyways suppose you strain it so much and now if you let go of the d-forming force now the material will not snap back to its original length there will be some permanent deformation left over here so imagine over here look at the steel imagine this is where we are right now this is the amount of strain that we have right now and if we let go of that d-forming force the material will not snap back to its original length now there will be some permanent deformation going on some permanent deformation left over here so once you go beyond point B you have permanently deformed it at least a little bit now if you look at this region this is a really interesting region because notice as you more pointer along over here along this point notice that the stress is pretty much a constant it's not changing stress is pretty much a constant but look at the strain the strain is increasing like anything all right which means we're keeping our poop pretty much the same but our material keeps on elongating keeps on elongating this region where the stress is pretty much a constant but the strain keeps increasing is called as the plastic flow region because it's it's behaving like a liquid it's sort of like flowing because the stress is pretty much a constant over here the topmost point of this graph the the highest stress that we can get who will call it as C and this point is called the ultimate ultimate tensile strength ultimate tensile strength it's called ultimate because this is the maximum stress your material can handle without getting seriously damaged so if you don't want to damage your material don't go beyond this point but this is just a test material we don't care what happens to it we want to go beyond this point and see what happens well if we go beyond this point we're gonna damage your material but notice something funny happens notice that the graph is curving down that means the stress is actually decreasing yeah the stream keeps increasing and what's going on I mean we're we we can real decreasing the pool and yet the strain is increasing and so for quite some time I was not sure what was going on over here so I went to my mechanical engineer friend who has actually conducted this experiment in his lab and I asked him like how is it that when you go beyond this point when you decrease the stress the strain keeps increasing I mean what's going on and so he explained to me that what's really happening beyond this point is that so imagine so imagine we are over here let's say we're over here so imagine we are over here right now let's say this is the ultimate tensile strength and if we go beyond this point what really happens is that there will be a small neck that is good that gets developed over here your material will start pinching like this and you may have seen this in rubber band when you stretch it too much and once the material pinches the cross-sectional area over here decreases and so if you calculate the stress in that region the stress is actually increasing even though you decrease the pull the stress in that region is increasing because of the smaller cross-sectional area but when we calculate the stress over here we always do that force per this cross-sectional area right because we don't know what this is and so once the pinching starts the stress in this region is increasing and from this point onwards all the strain will be localized in this region only this part will undergo stream and so even after decreasing the pull even if you decrease the overall stress the stress in this region keeps increasing and as the material elongates you will see that this neck becomes smaller and narrower and narrower and narrower stress keeps increasing a lot in this region eventually the material snaps the material breaks and that's at this point we'll call that as point D and that will be fracture fracture point so this is what steel experiences as we go beyond the elastic regions all the way to fracture point amazing isn't it and as told earlier different materials will have very different experiences and so what we'll do one last he will do is we'll play with this graph a little bit more imagine we had another material which had the linear region somewhat like this what could be what could we say about this material in comparison to steel well notice that it has a lower slope now but what does slope in this region tell us well let's see if we calculate the slope if you draw a right angle triangle then the slope is calculated as this side which is the change in the stress divided by this side which is the change in strain now what do you get when you do change in stress divided by change in strain you end up with the Youngs modulus can you see that therefore slope one way to think about slope is that it tells us how much the Youngs modulus is or it tells us how elastic that material is so for example if we said that aluminium for example aluminium had this graph then you could easily say that yeah aluminium has a lower Young's modulus than steel it is less elastic than steel but another way to think about this slope is you can also say that look if you compare aluminium with steel for the same strain notice for the same strain a lower stress is needed for aluminium right and therefore it's easier to stretch aluminium it's easier to deform aluminium compared to steel that's another thing that the slope tells us how easy it is or how readily the material deforms under a given stress another important number is this the ultimate tensile strength for example for steel it turns out that that number is about half a Giga Pascal but if you take something like say diamond one of the strongest material that is this number is close to about 50 Giga Pascal's I mean that's incredibly so diamond is incredibly strong so this is also a number that's very useful for engineers to understand how strong a material is one more interesting region is this one the region in which the stress is pretty much a constant but the strain keeps increasing the plastic flow region if you take a material or metal like say gold then it turns out that it has an incredibly long region over here it's pretty long what does that tell us about gold for example what does it tell us well that tells us that gold can be deep a lot in this region before it hits fracture in fact it turns out that if you take a gram of gold then you can stretch it up to about two and a half kilometers before you reach this point of fracture so such materials which have very high plastic flow regions where C and D point are very far apart we call him as ductile materials very duct as a goal is very ductile platinum also turns out to be extremely ductile on the other hand there are some materials which are exactly opposite of that meaning they have a very narrow region of the plastic flow in these regions the Point C and point D and point B all are very close to each other and I'm not exaggerating over here some materials are like this we call them as brittle materials I mean think about it glass is one example over here it's brittle because if you go a little bit beyond the elastic region like you go beyond the elastic limit then there's not much room left very quickly the material will snap very quickly it'll break and that's what glass does now here's the mind-blowing thing whenever someone says the word brittle what usually comes to my head is a picture of glass shattering which makes me feel like brittle is the same as weak but guess what brittle is not weak something can be extremely strong having a very high tensile strength but can be brittle at the same time for example diamond is brittle whenever someone says diamond is brittle it sometimes confuses people I should say well I thought diamond was our strongest thing right well guess what diamond is very strong it has an extremely high tensile strength but it's extremely brittle as in you can't elongate it so to put another way diamond is very strong it's extremely hard to break it but you can't stretch it into thin wires but I guess you already knew that