Stress & strain

when we explore elasticity of materials the two most commonly used terms are stress and strain so let's see what they are and why they are necessary all right let's begin it's something that we might be familiar with Springs we already know that if you take a spring of length L and let's say you stretch it by some amount stretch it by some amount and let go of it it tries to snap back which means there is a force trying to restore the spring back to its original shape we call that force as the restoring force and suppose that stretching that we produce the change in length is Delta L and let's say that the restoring force that's trying to bring the spring back to its original shape let's call that as F R our sense for restoring then there is a relationship between the two we call it as Hookes law and that relationship that Hookes law says that the restoring force is proportional to the displacement and we only look at the magnitudes over here we don't care about the directions because strictly speaking the restoring force is in the opposite direction of the displacement if you see properly but anyways the proportionality sign is just saying that if you increase the displacement you stretch it more the restoring force also increases makes sense if you think about it right and we've spoken about this in previous videos so if you require more clarity on this it'd be great to watch that video first and then come back over here but now the big question that we have the rumor tackle is can we generalize this concept to other things not just Springs because we've seen things like rubber band or steel bone rocks those are all elastic materials as well so can we extend the Hookes law to them that's the big question all right so let's say we have a rubber band we have a thin rubber band and imagine we stretch it so let's say we stretch it by point one centimeters the number is only for representation don't take it too literally and if we let go of it so if imagine we let go of this thing over here we know again the rubberband is gonna snap back which means there is a restoring force acting over here as well a restoring force again fr so can we say over here again restoring force is proportional to the amount of displacement just like for Springs can we do that that's the question well we could but answer is not very useful to talk about restoring forces in such situations and here's the reason why imagine instead of having a pin rubberband like this we had let's say a thicker rubber band very thick rubber band five times thicker rubber band but the same length and imagine we stretch it by the same amount and the same material then I'm pretty sure you're you agree with me in this case that your hands would be very shaky over here well because now we have much thicker rubber band we have five times thicker rubber band so I'm pretty sure you can visualize it'll be much harder to maintain this position well the reason for that is because now you have a much higher restoring force acting over here how much higher well if you think about it we can imagine this thick rubber band to be made up of five thin rubber bands right we can do that and since each rubber band is stretched by the same amount it produces the same restoring force so the total restoring force would be five times higher and that's why it'll be much more difficult to maintain this position so it means that imagine the restoring force over here was hundred Newtons as an example then over here the restoring force over here the restoring force would be five hundred Newtons five hundred five times more so notice that even though the stretching is the same the restoring force has changed and it depends on the thickness or in other words the area of cross-section right that's what that's what matters over here notice that the area were here was tiny let's say the area was one one meter square or whatever and then the area the cross-sectional area over here is five times more it will be five area so as the area increased the restoring force also increases so can we come up with a quantity that is independent of the so that it doesn't depend on the thickness and the answer is yes if you take the restoring force and divide by the area notice that number is the same in both the cases one hundred by one is one hundred five hundred by five is also one hundred and suppose you had let's say three times thicker rubber band then the restoring force would be three times as much but the area would also be three times as much and again if you do restoring force per area you again end up with hundred so a useful quantity over here would be not restoring force but the restoring force per unit area that's what's useful over here restoring force per unit area and we give this quantity a name and we call that as stress so this quantity over here is called as stress and it has a unit off let's see force per area so it'll be mutants per area is in meter square so Newton's per meter squared and what does stress tell us well just like how the restoring force in Springs tell us how hard the spring is trying to snap back to its original shape stress in general tells us how much how hard any material is trying to snap back to its original shape so notice that in these two cases although the restoring forces are different the stress is the same and therefore both these rubber bands will pretty much snap back in almost the same time and you might be wondering Oh why's that because there's more restoring force right shouldn't it snap back much quicker oh the answer is no because it's five times thicker as well if you think in terms of each rubber band fire our bands are there I mean if you think of it that way then each rubber band is again putting the same restoring force it's doing the same thing as before so restoring force could be misleading in thinking when it comes to thinking about snapping back but so so a more useful thing or the more accurate one would be to think about stress all right the next question could be what does this stress depend on could we say that this stress is proportional to the amount of stretching so if you had to say stretch this rubberband by 0.5 centimeters we would have 5 times restore during force and so five times more stress so could we just say stress is proportional delta L well turns out we can't do that and here's the reason why imagine this time we had a five times longer rubber band so let's say this was one centimeter long it's a relaxed length of one centimeter long and now we have a five centimeter long rubber band same material same thickness and let's say that we stretch it by 0.5 centimeters five times more stretching do you think they'll be guilty will be more restoring force now I want you to pause the video and think about this for a while all right the answer is no the restoring force will be exactly the same and here's the way to think about this you see since we have five centimeter long rubber band we could now assume that this is made of five such thin rubber bands and since we have stretched by 0.5 centimeter each of those rubber band have gotten stretched by 0.1 centimeter the same as before so do you see that each band is still getting the same restoring force of hundred Newtons in other words the stress because the area is the same the stress now is the same as before so just by looking at this number we can't conclude what happens to stress the says the stress is the same because if you take every one centimeter wire over here so notice if you take every one centimeter wire or the band of every one centimeter that's got stretched by the same amount so to increase the stress we need to increase the amount of stretching that's happening per centimeter not the total stretching that's all what matters so if we could increase that number then the stress would increase so the stress depends on how much stretching is happening per centimeter or in other words how much you're stretching per unit length that's what matters over here all right so the quantity that matters to us would be not Delta L but Delta L divided by L so amount of stretching per centimeter or per unit length that's what matters over here so notice in these two cases this number is the same point five by five is 0.14 one by one it also point one and that's why the stress on and the restoring force is the same in both of these and so what stress really depends on is not the change in length but the change in length per unit length and this quantity this quantity we call it as strain strain and notice it is unitless it has no units because centimeter and centimeter cancels so unit less quantity we also call this as the relative change sometimes and sometimes we also multiply by 100 and think in terms of percentage change for example notice 0.1 by 1 is 0.1 and if you multiplied by 100 we get 10 so we could say there's a 10% strain over here there's a 10% strain over here as well so you can also think of it as stress depends on how much percentage changes happen in the length not the absolute value of the change in the length all right one last thing I want to do is help you remember these new quantities stress and strain I mean sometimes you might get confused which one is which so here's how I like to think about it so usually when exams are coming close by we get stressed from inside and that's how I remember the stress has something to do with something that gets generated inside and what gets generated inside is the restoring force and that's how I remember that stress is related to the restoring force and similarly when when you lift something it left something very heavy chances are that your muscle to get pulled and we usually say that your muscles are being strained and that's how I remember that strain has something to do with something being pulled or change in length