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Shear modulus

Let's explore a new modulus of elasticity called shear modulus (rigidity modulus). This will also explain why our bones are strong and yet can be fractured easily. Created by Mahesh Shenoy.

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Video transcript

the strongest born in the human body the femur the thigh bone is so strong it can withstand at least six to seven times your own body weight and yet take a fall from your chair just the right way and you can easily break that bone how can something be so strong and yet so weak at the same time we have seen in previous videos that if you have an object which is fixed at one end like a building which is fixed to the ground and if a force is acted upon on the other end then to find the effect of this force we imagine that this body is made up of a lot of planes which are parallel to that force and what the force does is makes this plane slide past each other so all the planes end up sliding past each other and as a result the building deforms and we call this deformation as shearing and the material fights back by generating a restoring force and if we calculate that restoring force per unit area we call that as shear stress and the distance that one plane slides with respect to another which are a unit distance apart we call that as the shear strain and we've spoken a lot about this in previous videos and so if you are not familiar with this or you require a refresher it would be a great idea to go back and watch that video first and then come back over here in this video we're going to explore the connection between shear stress and shear strain now you may have already learned about Hookes law which states that within the elastic limits stress is proportional to strain and guess what that same Hookes law even works for shearing so let's go ahead and write down Hookes law for shearing we could right now that shear stress shear stress is proportional proportional to shear strain shear strain and this just says the more strain you produce the more deformation you give to the body the more stress it will generate to restore itself back and now the proportionality can be replaced by a constant which we call as the modulus of elasticity and that modulus of elasticity over here for sheering the Elector we use for that is G and we call that as shear modulus shear modulus it's also called as rigidity modulus rigidity modulus everyone we learn a little while why it's called so all right so let's get some feelings for what the shear modulus is telling us first of all the units of shear modulus notice that the strain has no units just like any strain it is unitless quantity so the modulus of elasticity must have the same units as the shear stress right and again this is true for all moduli of elasticity they have the same units as stress and so the units would be Newtons per meter square Newton's per meter square and now let's see what this value of G is telling us all right if G is very high if G is very high what does that tell us well that tells us that for a very small shear strain like a strain like this deformation the shear stress produced the restoring force per unit area produced would be very high which means the material is very elastic it tries to snap back very quickly but it's also telling us that it's extremely hard to deform it right because if the stress produced is very high it becomes very difficult to deform it so in other words we could say that if G is very high the body tries to maintain its shape it's difficult to deform it it is behaving more rigid right now what's the meaning of the word rigid rigid means a body which is able to maintain its shape that's the idea behind rigidity so higher the value of G more rigid that body is and that's why G is called as the rigidity modulus so we could say that the body is um highly rigid highly rigid and then you say the word rigid what should come to your mind is maintains shape maintains shape or resist very strongly to changes in shape and if we go extreme and if G goes to infinity then it means it's a perfect rigid body right I mean even for a very tiny amount of stress strain the body would produce infinite stress so it would be impossible to deform that body on the other hand if G is very low if G is very low then it is less rigid it's easier to deform it it resists less to changes in deformation and now if you follow this chain of thought the question is what what would happen if G was zero I want you to pause the video for a while and just think about its implication if we say that a material has zero g value what would it mean what can you understand about the material all right it would be exactly opposite of being rigid if G was zero it means it's not going to resist changes in deformation at all now think about it if this material suddenly had zero value of G then even for a very tiny amount of force that I put over here the layers would just keep sliding past each other forever and as it keeps doing that eventually the whole but the whole building my deform and start flowing that makes sense so if G is equal to zero it's telling us that the material starts flowing in other words we are talking about fluids so fluids have zero shear modulus all right so this is for fluids now fluids are liquids or gases and their main property is that they flow they don't have a definite shape now the same thing can be said in another way we could just say they have zero shear modulus it would mean the same thing and similarly solids in contrast don't flow and they have a definite shape again that same thing could be said by saying that they have a nonzero shear modulus so next time if someone asks you what's the difference between a key difference between solids and liquids or you can tell them the usual answer that solids maintain their shape and liquids don't liquids for or you can say the same thing in a very technical intimidating sciency way you could tell them liquids have zero shear modulus but solids they have nonzero ones and people would be like whoa all right one last thing I want to talk about is shear strength so let me make a little bit space over here so shear strength will remember that Hookes law only works within the elastic limits which means if you strain it or if you deform it too much then the then the deformation could even be permanent and if the strain is extremely high if you deform it a lot then the chances are that the stress generated could even break the material you know that right everybody keeps saying too much stress is not good for your body it's the same thing applies here as well so strength is just a measure of what's the maximum stress a material can handle without breaking itself so compressive strength for example tells us what's the maximum compressive stress a material can withstand without breaking itself tensile strength for example would be what's the maximum tensile stress a material can withstand without breaking itself so what do you think shear strength is or shear strength is the maximum shear stress maximum shear stress about which the material will fracture so maximum shear stress the material can handle without breaking itself so the modulus tells us how elastic the material is how quickly the material tries to snap back how rigid it behaves but the strength just tells us the maximum limit you go beyond that the material will fracture and so strength is another thing that engineers should take care about when they're designing things and finally one natural thing we might want to do is to connect the properties of elasticity in shearing with tension and compression remember that the elastic modulus for tension and compression is the Youngs modulus so a natural question could be is there any connection between shear modulus and young's modulus the turns out that there is a very complicated relationship between them we're not gonna worry too much about that but when you simplify it and you apply to many material it just happens that the shear modulus happens to be pretty much pretty much one-third of Young's modulus which tells us that most materials are less elastic when it comes to shearing compared to compressing or stretching them Penza but things get really interesting when we start talking about strength and this is the last thing all right because strength talks about when the material breaks guess what materials also have different shear strength tensile strength any one different compressive strength and that brings us back to the question of bones it turns out that our bones have very high compressive strength and this is why you can stand for a long time or you can jog you can pretty much jump you can carry a lot of weight because in during all these times you're actually compressing your bones and our bones won't fracture so easily under compression but guess what it turns out that they have a very low shear strength meaning it's much easier for the bone to get fractured when you shear it and these kinds of forces are born experiences when they get a sharp blow like when you take a fall and it's for that reason sometimes you can just have a moderate fall it doesn't feel like it's a big thing but you may have accidentally sheared that bone so there's a good chance you may have fractured it and so please remember for future that if you ever take a fall and it doesn't look like it's a big deal from outside there's a still a chance that you could have sheared the bone and because the bone has a very low shear strength there are all the chances of fracturing it and so always get it checked