In this video we're going to derive an equation for the transverse tensile modulus of a unidirectional composite. So again we have our block of composite, we have fibers running through it, these farmers are continuous, parallel they're the same all the way along, but this time we're interested in the properties perpendicular to the fibers, this is the transverse direction. Now to do this we first of all have to make quite a bigger assumption about the structure of the composite. So imagine taking a cross section of the material and having a look inside. What we've got is we have fibers surrounded by matrix. Now in order to do this calculation we have to assume that instead of having separate fibers, the fibers are in fact arranged all in layers. So instead of discrete fibers we have layers of the fiber material going all the way across the composite. And in fact we have to make an even bigger assumption, we have to assume that all of these layers are then squashed down into one layer which occupies the whole bottom half of the composite. Now this isn't a great assumption, so the numbers that come out of this calculation aren't as good as the longitudinal direction, but they've got enough for understanding what's going on. The reason we make this assumption is if you imagine now we'll look back side on, if you imagine the composite with all the fibers squashed together and then the matrix on top, so this is fiber, this is matrix. Now if we apply our load all of the load has to go through the matrix and then all of the load has to go through the fibers. So in fact the load in the composite in this case is equal to the load in the matrix is equal to the load in the fibers. And because of this assumption here assuming they're slabs they all have the same cross sectional area. So the load is the same, the area is the same, so we can say the stress is the same. And this leads us into the main assumption for this calculation which is the ISO stress assumption. With longitudinal it is ISO strain, now it's ISO stress. We're assuming that the composite has a stress which is the same as the stress in the fibers which is the same as the stress in the matrix. And to write that algebraically we have sigma for stress, sigma c for the composite is equal to the stress in the fibers is equal to the stress in the matrix. That's the ISO stress assumption. Now the thing we have to do is consider the sample being elongated. So here's our original sample and this has a total thickness of Tc, thickness of the composite. The thickness of the matrix layer is TM, the thickness of the fiber layer is TF. Now we're going to consider elongating this, we've applied a force and this thing has got longer. Now the matrix and the fiber are going to elongate by different amounts because they have different moduli. But that doesn't really matter to us, it doesn't matter for this analysis. What we're going to do is we're going to express the new thickness as T plus a delta T, a change in thickness. So if this was TM before, now it's TM plus delta Tm, there's a change in thickness. If this was TF before, this is TF plus delta TF, the little delta just means changing. And if this was Tc before, now it's Tc plus delta Tc. Now it should be pretty obvious that in this case, Tc was TM plus TF. In this second case here, it now should also be obvious that the change in thickness delta Tc is equal to the sum of these changes. So previously, the thickness of the composite was the thickness of the matrix plus the thickness of the fibers. In this new elongated state, the change in thickness of the composite is equal to the change in thickness of the matrix plus the change in thickness of the fibers. Now we don't like talking about change in thickness, we prefer to talk in strain. So remember the definition of strain, strain is just change in length over original length. So we can use the terminology we've introduced here because of thickness, this is the same as change in thickness over thickness. We can rewrite this as delta T equals strain times the original thickness. Change in thickness is equal to the strain times the original thickness. This is just rearranging the definition of strain. So we can actually write out three different versions of this. When you increase the date of the sucking of the thermo-b乘. until the specific fineness of the dial nor the番 – with the body of the pessoa, you can really ruin the fit. On top of this, the secondton's will be the movement of theичего that Forfluen thinks it's spacemen. So basically, bywarzmann bureauy means the thickness of the composite is the strain in the composite times the thickness of the composite is equal to the strain in the matrix times the thickness of the fibers. Now we can take these and we can substitute them into here and then we can stop talking about changing thicknesses and we can start talking about strain. So just making that substitution that gives us strain in the composite, tells us by thickness of the composite equals strain in the matrix, tells us by thickness of the matrix plus strain in the fibers, tells us by thickness of the fibers. Now we don't want to be concerned with the actual thickness of the composite material. But in the same way we didn't want to be concerned with the actual cross sectional area in the longitudinal case. So we're going to play the same trick. If we divide through by the thickness of the composite, we end up with strain in the composite equals strain in the matrix, tells us by thickness of the matrix divided by thickness of the composite plus strain in the fibers divided by thickness of the fibers divided by thickness of the composite. We can think of this now and this as thickness fractions. So fraction of the total thickness does that layer occupy, this is between naught and 1. And because of the assumption we made right at the start, that there's only two layers and the composite is the same, no matter what cross section we take, these things are equal to the volume fractions we talked about previously. So we can just directly make a substitution and substitute these thickness fractions for the volume fraction. So that's the volume fraction of matrix and that's the volume fraction of fibers. So we don't want to really be thinking about strain, we're trying to get to an equation about modulus. So now would be a good time to bring in the definition of modulus. So the definition of Young's modulus is E, the modulus is equal to stress divided by strain. And we can just rearrange that in terms of strain because it's the strain we're trying to get rid of. So now what we do, because we want to get rid of the strain, is substitute all of the strains in this equation for this definition of Young's modulus. So that gives us stress in the composite divided by modulus of the composite equals stress in the matrix divided by modulus of the matrix times by the volume fraction of matrix that hasn't changed. Plus stress in the fibers divided by modulus of the fibers times by the volume fraction of fibers. Okay, so just to recap, all we've done there is take this equation and for each value of strain we've used the definition of Young's modulus to substitute for stress and modulus. Now right back at the start, we made an assumption about stress. We said we were in the iso stress condition, which means all of these stresses are equal. So just like we did in the longitudinal case, these three terms are all equal so we can essentially divide by that value and cancel them out. Which gives us one divided by the modulus of the composite equals one divided by the modulus of the matrix times by the volume fraction of matrix plus one divided by the modulus of fibers times by the volume fraction of fibers. And that's what we were trying to get to. That is our equation for the transverse modulus of the unidirectional composite. And of course you could just express this in terms of all in terms of the volume fraction of fibers. That's just removing the M and replacing it with one minus VF and I've just done a little bit of tidying up by putting them on top. That is an equally valid expression.