[SOUND] Hi. This is Module 38 of Mechanics of Materials part one. These learning outcomes are to define Isotropic materials, and to define or develop Generalized Hooke's Law for Isotropic Materials. So we callback to earlier, near beginning of the course when we came up with the stress-strain diagram. And for material properties, we said that the stiffness in this linear region was equal to or the slope was equal to Young's Modulus, or the modulus of elasticity. And in that linear elastic region, we had Hooke's law. And this was for a torsion test, where the stress was equal to Young's Modulus times the strain epsilon. And so this was for uniaxial loading, but we can extend this for use in more general engineering applications, for situations involving biaxial or triaxial loading. So recall Hooke's Law, it assumed elastic behavior. But let's add another common assumption, that the material is isotropic. And isotropic is defined as having the same material properties in all directions. And that means that Young's modulus is the same whether we pull in the x direction, the y direction or the z direction. And the Poisson's ratio is the same in the xy and the z directions. And so some isotropic material examples are rubber or steel, most metals. And so here's an isotropic material, a rubbery, elastic type material, whether I pull in this direction or I pull in any other direction, the material properties are the same, so that's isotropic. As far as anisotropic, I've talked briefly about them before in the course, and some examples would be carbon fiber or wood. Here's an example of wood, you can see that wood, as you know, has grains, and so If stress this in one direction versus a different direction we're going to have different material properties because of the grain. Here's also a tube of carbon fiber and you can see that the fiber runs along in diagonals. And so if we Go ahead and apply stresses to this element or this engineering element. We can get different material properties in different directions. And so that's the difference and so what we're developing for Hooke's Law is for a generalized for isotropic material. And so, I want you to now Recall Poisson's Ratio, which we talked about earlier in the course. And we defined Poisson's Ratio, with the Lateral Strain and the Longitudinal Strain. And the Poisson's Ratio was negative the lateral strain divided by the longitudinal strain. And so we're now going to consider biaxial principal stresses. And so let's first stress it in the x direction. And so for the stress in the x direction, Poisson's ratio says the lateral strain will be the strain in the y direction, and the longitudinal strain is in the x direction. And this is the expression. Epsilon x is by Hooke's Law sigma sub x distress in the x direction divided by Young's modules. Similarly, epsilon sub y is going to be minus Poisson's ratio times epsilon sub x from this equation here. And so I can substitute in for epsilon sub x using Hook's Law and we get epsilons to y is minus Poisson's Ratio times the stress in the x direction divided by Young's Modulus. We can do a similar approach when I add biaxial stress in the y direction. So I have a biaxial stress condition. Here now the lateral strain is the strain in the x direction and longitudinal strain is in the y direction. and we know that epsilon sub y by Poisson's ratio is the stress in the y direction divided by Young's modules. And the strain in the x direction is equal to minus Poisson's ratio times epsilon sub x or epsilon sub y. I can substitute Young's modulus and I get this expression. So now let's take the x direction and the y direction applying principle stresses, let's put them together. And so here's what we came up with so far. If we combine those, we're going to have a biaxial loading condition. And for epsilon sub x, we're going to get sigma sub x over Young's modulus for the Hooke's law strain in that direction. But then we're also going to have to subtract out the Poisson's ratio effect from biaxial loading in the y direction and that's shown here. And I can do the same thing for epsilon sub y, it's equal to by Young's module sigma sub y over e, but I've got to subtract out the Poisson's effect due to the stresses in the x direction and we get this result here. And so that's a combination of loading in both of the directions. And so let's now take those equations, multiply the first equation by e, or Young's Modulus. And multiply the second equation by e and Poisson's ratio. And then we're going to add them together and when I do that, you'll see that I come up with this. On the left-hand side I have epsilon of x times Young's module plus Poisson's ratio times epsilon sub y times Young's modulus. And then I have sigma sub x on the right hand side. The minus Poisson's ratio times sigma sub y, cancelled with a plus sigma sub y in this equation, when I add them together. And I end up with minus then just the Poisson's rato squared times sigma sub x. And so, here I've factored out the sigma sub x on the right-hand side, carried it to the left-hand side by factored out Young's modulus for this side. And generalized Hooke's law for biaxial stress-strain then for isotropic materials is shown here. And so now we have strains on the right-hand side. And stress is on the left hand side. So relationship of stress and strain for a biaxial loading condition. And you can do the same sort of approach for the stress strain relationship in the y direction for biaxial loading, and this is what you get. And so here is the results. This development, you'll notice I did for just principle stress loading with only normal stresses in the x and the y direction. However you'll recall back that for small deformations normal strains are unaffected by displacements perpendicular to the normal strain such as those produced by shear strains. And so therefore, these equations that I've developed are equally valid even when see your stresses are exist on my stress blocked. And so, you can even extend this, this type of approached for a try acts to your states of stress and let you look that up and references mechanics some materials books. But, we're going to limit ourselves in this course to just biaxial stress-strain relationships, and again we're limiting this however for isotropic materials but it's a generalized Hooke's Law. And so now when I measured strains I can find the in plane shear strains and normal strains. And I can also find the stresses, and so very valuable tools. And we'll see you next time. [MUSIC]
