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Shear stress and strain

How do we really tear paper? This seems like a silly question, but the answer is 'we shear it'. So let's explore this concept of shearing. Created by Mahesh Shenoy.

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Video transcript

this paper has a very nice tensile strength hmm it also has a great compressive strength Wow this paper must be really strong right what you just saw could be an engineer's nightmare you could build something which is extremely strong which can easily withstand compression and tension but it could break very easily under a different kind of stress so what is this new kind of stress this seems quite important from the structural point of view right well this new kind of stress that we're going to talk about in this video is called shearing and by the end of this video we would have familiarize ourselves with this concept of shearing I think one of the best examples to understand shearing is by taking a book if you take a thick book like this again it's very strong under compression and tension it's extremely difficult to distort it or deform it by compressing it or by pulling it apart but notice just with my fingers I can easily deform it in a different way notice that my fingers over here are putting a force parallel to the surface of the book this way and as a result the papers are sliding past each other this sliding of papers past each other is what we call as shearing all right let's look at it in detail now imagine we have a cylinder a cylinder like this which is stuck to the ground let's say here's the ground we've stuck it over here this way and imagine we put a force we put a force on the cylinder parallel to the cross-sectional area just like what we did the book like this now just like how the book was made up of a lot of pages we could assume that this cylinder is made up of lot of cross-sectional planes like this that's the trick to understanding shearing always assume then whatever object you're dealing with is made up of a lot of planes and make sure those planes are parallel to the force that is acting or the force that's trying to deform the cylinder so imagine a lot of like this so due to this force as we saw just like with the book the cylinder will deform this way so it was like this before now it's like this somewhat like this all right and all the planes have slid past each other just like the papers did in that book now here's one question why does it stop why does this sliding thing stop why does it keep on sliding forever well what must be happening is that the material must be trying to restore it back so there must be a restoring force acting over here but what direction is the restoring force acting well if we concentrate on a couple of planes it will be easy to understand so let's say we zoom into a couple of planes let's say a topmost plane and the plane right below that if we zoom into that let's say so here is a plane a little bit below this so this is the plane a little bit at the bottom so this one over here and let me draw another plane the topmost plane over here let me just put some gap between them so it's easier to easier to visualize this all right so we're zooming in these are two planes this is the topmost one and this one is the one that's right below that and we are putting a force on the topmost plane this way and as a result this plane is sliding past this flame this way it's sliding past but it doesn't keep on sliding forever and the reason for that is as it slides because of the intermolecular forces this plane tries to pull it back so there must be a restoring force acting in the opposite direction this way and as the plane slides more and more the restoring force increases more and more eventually the restoring force equals the external force and then the whole plane comes into equilibrium and and that's how eventually the whole material comes into equilibrium now if we take this restoring force and divided by this area this cross-sectional area then we will call that kind of stress Remer stresses the restoring force per unit area so that kind of stress that stress is called as shearing stress shearing stress so shearing stress is this restoring force these restore this restoring force divided by the cross-sectional area so what should come to your mind when you hear the word shearing is Plains sliding Plains sliding past each other all right this should come to your mind past each other that's how I like to think about sharing now the next thing could be how do we calculate the deformation well since in shearing plains are sliding past each other we could just calculate how much one plane has slid past another for example if you take this topmost plane and compare how much it has slid past the bottom most plane the bottom was fun that is initially the topmost plane was over here here is the bottom was plane it has slide it has slid from here to here so we could say this is the amount of deformation Delta X but guess what if you take any other plane so for example if you take this plane then it has only deformed or it is only slid by slip past by this much amount so Delta X itself is not a good measure because different planes slide by different amount a better way to measure the deformation would be to calculate how much one plane has slid past another plane which are a unit distance apart let's not just take any two random planes but only take the ones which are a unit distance apart that would be a great way so imagine this length or the distance between this and this plane let's call that as L that would be the length or the initial length of the cylinder then from this we could say the two planes which are L distance apart have slid past by an amount Delta X so if you take any two planes which are a unit distance apart how much have this slit past each other well they could just do a cross multiplication and we could get this as Delta X / L this tells us how much - planes which are a unit distance apart have slid past each other and this is what we call as shear strain or shearing strain and notice you can also calculate this number for these two planes you'll get the same answer notice that if you take these two planes this would be Delta X this would be L and this divided by this is the same as this divided by this because they are similar triangles another way to think about this could be by taking this angle if we call this as the angle of shear or the angle by which this thing has twisted then notice that if you take this triangle or this triangle then the ratio turns out to be the opposite side divided by the adjacent side opposite side by the adjacent side so we could just call this as tan of the shearing angle well one last thing we'll do to understand shearing perfectly is we'll understand major differences between shearing and tension and compression so if you wanted to say compress this particular this particular rod then you would have pushed it this way maybe this would be the push they would have done now comes the question what would these planes be doing under the compression what would happen to them well let's just shift this a little a little bit towards the right let's make a little bit of room I'm running out of place over here all right so again if we draw these planes so imagine that's over here it's a little crowded please excuse the drawing but if I draw the bottom plane over here and the top plane over here over here and you're pushing down on it then what would happen to the plane is they would they would get pushed close to each other and as a result there would be a restoring force acting this way this way so the major difference that you see between compression and shearing is that in compression the planes come close to each other and maybe in tension the planes go farther away from each other but it's shearing they don't do that in shearing they just slide past each other and notice if you look at the restoring force the restoring force is perpendicular to the area even when it's come to tension if you had tension over here the restoring force would be downward again perpendicular to the area but notice over here the restoring force is parallel to the area right that is the major difference that you see between shearing and the compression and tension so the restoring force over here is parallel parallel to area now when I try to tear this paper notice that the way I put the force is this way and this way now consider a plane which is parallel to this force so it will be something something like this parallel to this force this way and this plane will run along I'll run along this length and it will have a very tiny R it and that R it is so tiny because the thickness is so small so the plane would be a rectangular one shown like this will be like this and the area of this plane is incredibly tiny and that's all because it has a very tiny Verret and because this area is so tiny even a modest force is going to produce a huge shear stress on this paper the stress is so huge that the paper can't handle it and that's why it just tears alright let's end with what really fascinates me about this I used to always think that scissors would cut paper this way but guess what they don't do it that way instead they cut it this way the two prongs are not aligned as you can see here that's the key this is how a scissor puts a huge sheer stress on the paper just like how we do it when we are tearing it and ends up cutting the paper so next time you are using a scissor to cut a paper remember you are shearing it