 So, we've introduced the idea of angle of friction as a sort of material specific parameter. So we know clays have a certain angle of friction and cohesion and sands have a certain angle of friction and cohesion. But actually, it's not a material specific parameter. So we've introduced the idea of angle of friction and cohesion, the shear strength as sort of material specific parameters. But they're not really. Angle of friction, like this graph shows, changes of density with the material. So with increasing relative density, we can also see that the angle of friction increases. And this graph also helps to define angle of repose, which is the angle of friction at the lowest density. So angle of friction changes with changes in density. We know moisture content or water content of the soil affects the cohesion. So these aren't really material specific parameters. And what would be really useful in soil mechanics is to have a set of parameters that were specific for the type of material that we were looking at. So what critical state theory does is it models the soil at the point of failure as a frictional fluid, which means that it starts to flow as a almost like a liquid, is a bit like molten metal. The problem in some ways is that a lot of soils don't behave like that. So the use of critical state theory does come with substantial health warnings. So if we return back to this diagram where we when we introduced the shear box test, where we have the shear strain against the x-axis and specific volume against the y-axis, and what that showed was that when we shared our sample, the sample reached a point called the critical specific volume. So depending on whether we were talking about initially dense or initially loose material, it didn't really matter because it would reach the same critical specific volume. Now that critical specific volume is in is on the critical state line. So if we did the shear box test on different normal effective stresses, put a new stress onto the lid of the box and replicated the test, we would get different lines on this diagram for different normal stresses. So there would be a relationship which we could then plot between the normal effective stress and the critical specific volume. And that would generate a straight line, or theoretically generate a straight line. But we also know that there's a relationship between effective stress and shear stress within the material, the more cool on failure envelope. So what we can do is plot these three things together on one set of axes and that would be in three dimensions. So let's just make some space for that. So if we put those two lines together on the set of three axes where we have normal effective stress, shear stress and specific volume, we could see that it would form a line that looked like this. And the blue line underneath is the projection onto the normal effective stress and specific volume axis. And the yellow line here is the projection onto the shear stress specific volume axis. And the orange line is the critical state line. So what usually happens is this, instead of expressing the information like this, the information is expressed in terms of stress invariance. So if you remember that you have two two-dimensional stress invariance, you have t, which is the radius of that most circle. So you have sigma max minus sigma min divided by 2. And you also have s, which is the centre point of the circle, which is then sigma max, or the average stress. Those are the two-dimensional stress invariance. And those are appropriate when we're talking about conditions of plane strain. But when we're talking about conditions that aren't of plane strain, like in a triaxial test, we need to then bring in sigma 3, the intermediate principle stress. And there are two three-dimensional stress invariance. There's one called the deviatoric key, which is essentially saying similar things to t here. What the deviatoric stress is is the stress or the contribution of the stress that deforms the material. And we also have p, which is the average stress. So the reason why we use these stress invariance is that it lets us talk about generic stress conditions within a soil without referencing any specific absolutes. So that's why it's quite useful. So if we again do the same thing with, but now plotting q against p and specific volume against p, we can draw another two lines. So the first line is a straight line. And that has the equation q equals mp. And on this graph, we can draw another straight line. And that has the equation of v equals big gamma minus lambda line. So these are the critical state lines represented in this stress invariant space. But what it has been able to define is material-specific parameters. So m is not really material-specific parameter, as is gamma and lambda. So we've gone from a situation now where from using angular friction and cohesion, which are, which change depending on conditions within the soil. It's now talking about material-specific parameters. And that's really the benefit of critical state theory. So this also helps demonstrate the concept of stress paths. Now stress paths are the paths that the stress conditions within your soil go through during loading. And these change depending on whether you're looking at a drained or an undrained material. So for drained material, you might start off with a condition here where you have a given value of p. And in a drain test, you would increase the stress conditions until you reach the critical state line where the material would then flow or notionally flow along that critical state line. So that's for a drain test. And from that, you could then make some sort of statement about the strength of the material. But in terms of undrained conditions, the stress path might look completely different where it might start at the same value. But because water pressure isn't dissipating, you might have conditions that look something like this. Well again, it flows up the critical state line, but it reaches the critical state line much sooner than what would happen in a drain test. So you get, in this case, a much different value for the strength of the material. So that's another reason why this critical state theory is important. And it also helps explain why the stress paths to reach that critical state are also important to understand.