[MUSIC] Welcome to module 29 of Mechanics of Materials part one. Here are the topics that we've talked about in the course. We just finished up stress concentrations. Earlier in the course we discussed briefly normal strain and shear strain. We're going to look now at strain in more depth. And so today's learning outcome is to define what we'll call two-dimensional or plane strain. And so you'll recall here is a axial loaded member. When it's pulled on we get a strain. The strain is given the symbol epsilon. The normal strain. It's an elongation per unit length. And so it's dimensionless. The sign conventions are if we're intention which causes elongation which is a positive strain, compression causes a negative strain. We also talked about shear strain which is angular distortion or shear distortion due to shear stresses. Here's our stress block, shear stresses are being applied and we get shear strain. Which is an angle. That angle, gamma, is a addition of the angle gamma 1 and gamma 2, or pi over 2, 90 degrees minus this angle beta. And the sign convention was that a positive sheer stress is shown here causes a positive sheer strain and vice versa so that the angle is reduced on two positive or two negative faces. So, we find that measuring strain is often easier to measure than stress, in actual engineering [INAUDIBLE]. And we use experimental analysis techniques to measure those strains, and we're going to actually talk about that, how you do that, later in the course. After we have the strains, we can use stress-strains relationships that we'll also derive later to calculate stresses. We're going to focus on biaxial or two-dimensional loading. We're going to subject this small unrestrained rectangular parallel piped to a system of two dimensional loads, which is going to be a combination of axial and shear loads. And when we do that, this is what our parallel piped is going to look like. It's going to stretch out in the x direction, it's going to stretch out in the y direction, and it's going to experience a sheer distortion as well. And so you'll recall for the normal strains, this was our expression. And so if we talked about strain in the x direction, this is what it would look like. And so what we find is dx prime is our new length in the x direction. So down here we have dx prime. If I multiplied through by dx, I'd get equals dx, the original length, plus epsilon x dx or this is equal to 1 plus epsilon x times dx. And we can do a similar calculation for the change in length in the y direction. So on the y face. We're going to have dy prime, the new y length is equal to dy plus epsilon y dy, which is equal to 1 plus epsilon y d y. So that's our deformed block. Now let's take our rectangular parallel pipe and then let's say that we have no strains in the z direction. No normal strains nor no shear strains in the z direction. And what we call that is the case of plane strain. And so here is our distorted block with the developments of the lengths in the x and the y direction as I did previously. And so we have plane strain, no strains in the z direction. Recall now back to when we did plane stress. This was the condition of plane stress. For plane stress, all the out of plane stresses were equal to zero. And that's shown here. And we said that we had a small relative dimension in the z direction with no surface stress in the z direction. So there was no stresses in the z direction but there could be strains in the z direction. And the example that we used was thin plates, something like the panels of an aircraft wing. Now we're talking about plane strain. And in an analogous way there are no strains now in the z direction but there can be stresses in the z direction. And so we have generally have with a case of plane strain, a large relative dimension in the z direction, and we have restraints to prevent strain in the z direction, so therefore we can't have stresses in the z direction. And the example we use, or the types of problems we solved are what we call thick plates, or thick dimensions for our structures. And so, plane strain here again. Some examples would be things like dams, or retaining walls, or tunnels, where you have a large z component which is restrain, so you can get stresses in the z direction but no strains. Another example would be bars or tubes that are compressed by forces normal to their cross-section in the z direction so that they have no strain in the z direction. They have strains in the x and y direction, but this is plane strain. So the question becomes as an engineer, when do I use for my particular example, plane stress or plane strain. And you must use your engineering judgement in modeling. And beware of the assumptions that you make. And so, we'll go through and develop more of the plane strain case in future modules. [SOUND]