[MUSIC] Dear students from today's lecture on we are going to discuss the strengthening mechanisms of metals. So there are quite a few different strengthening mechanisms for metals and we'll start from the grain refinement strengthening mechanism. And this applies to polycrystals, right? And in polycrystals, we know that the plasticity or the ductility is accommodated by the synergistic and cooperative deformation of different grains, right? And usually plastic deformation of polycrystals is accommodated by multiple slip systems. So, because you know that you have multiple slip systems in the certain crystal structure, in FCC structure you have 12, right? And different grains in the polycrystal may have different Schmid factors and different slip systems may be activated. And in general, you can easily know that the grains with the highest Schmid factor will deform first because it cracks bonds to the lowest uniaxial yield strikes. And grains with low Schmid factor haven't really deformed at a time. So it's a non uniform deformation and this makes your deformation, plastic deformation of polycrystals more complex than that in single crystals. Because to maintain the cohesion of the materials, so the polycrystal is still a same piece of material under deformation, although the grains are deforming at different degrees. So to maintain the cohesion of the polycrystalline material, the grains has to cooperate and to deform cooperatively to fit the dimensional change of each of the neighboring grains. And this makes your plastic deformation of polycrystals much, much more complex than single crystals. Important idea of plastic deformation is that plastic deformation happens at constant one. So why is that? So suppose you have a sample with initial dimension of x0, y0, z0, and then your initial volume will be x0y0z0, right? And suppose after deformation your sample have a geometry or dimensional change, and your x0 becomes x0+dx, your y0 becomes y0+dy and z0 becomes z0+dz. And then your total volume becomes (x0+dx)(y0+dy)(z0+dz), okay? And then your relative difference between your final volume and your initial volume after plastic deformation can be calculated by comparing these two expressions, right? And one assumption is that, your second order product dx times dy, dx times dz = dy times dz = your third order product dxdydz becomes 0. So as long as you make this assumption, you can easily get this constant volume condition from the initial and final volume of your plastic deformation. So anyhow, you have this constant volume condition between your three normal string values, right? And because in general in three dimensional case you have three normal strengths, three shear strengths, and you have this constant volume condition, so that you only have five independent strength values. And if you consider one particular slip system will accommodate the plastic deformation of each of the independent strain parameters. And then there has to be five Independent or at least five independent slip system operating at the same time to accommodating the plastic deformation of these independently five strength values. In other words, those crystals with higher number of slip system are easier to deform. So that you know the FCC and BCC have fairly large number of independent slip systems. So FCC and BCC metals are easier for defamation as compared with HCP metals which usually only have three independent slip systems, okay? Because of the lattice distortion at the grain boundaries in polycrystals you have different crystallographic orientations in between the neighboring grains. So usually grain boundaries is the obstacle for dislocations, right? So, this is your grain boundary strengthening mechanism. So suppose you have a single grain here, which is a neighbor of several other grains with different crystallographic orientations. So suppose you have this array of dislocations trying to propagating across the grain boundary. What will happen is usually that because you have a discontinuous lattice at a grain boundary, so that your dislocation array under this shear stress will transmit across your grain boundary. And they will pile up and they will block out the grain boundary. So that the dislocation motion is prevented further and that makes your material stronger. And this is called your grain boundary strengthening or grain refinements strengthening. Because if you make your grain size smaller you will have a larger number of these grain boundaries per unit volume of your material so that you will have more effect of this grain boundary block of your dislocations. And this famous Hall-Petch relation is proposed in the 1950s to describe this grain refine and the strengthening. Where your, strings is the summation of a constant term plus a prefactor k times the inverse square root of your grain size. So again, when your grain size is smaller, you will have a larger yield strengths. So that is the grain refinements strengthening, okay? So again, the yield strains is this term, the yield strength is correlated with the yield strength of a single crystal, sigma zero and the material related coefficient, k here, so the k is prefactor here. And k really depends on what particular type of material you're working on. And d is the grain size. So the smaller the grain size, the higher the yield strength according to this Hall-Petch relation. So here are again two comments, Hall-Petch relations. It turns out that the Hall-Petch relation not only applies to poly crystalline metals, but it but it also generally valid for the strings of multi layered materials for the fracture strengths of brittle ceramic materials. And this part will be covered later when we talk about the mechanical behavior of ceramics. Another common is that unlike other strengthening mechanisms that we are going to talk about in subsequent lectures, in general, grain boundary refinements strengthening doesn't reduce the materials ductility and toughness. So the overall mechanical properties can be improved. Okay, so it's a very good method. And a final comment is that what we have just seen is mainly applicable for relatively low temperatures. And at these low temperatures, your green boundary have sufficient strength to resist dislocation propagation across the grain boundaries. Or equivalently, your grain boundary strengths is larger than your grain interior strengths. And this is the case for relatively low temperatures. However, at relatively high temperatures or particularly at temperatures that are higher than one half of your melting temperature, the green boundary will have sufficient thermal energy to move by themselves. And in those cases, grain boundaries may no longer become obstacles for dislocation motion. And the presence of grain boundaries will actually reduce the strengths of your material. So essentially, you can imagine that there has to be a so called Iso-strength temperature, where your grain boundary strength is equal to, so this is your grain boundary strength, your grain boundary strength is equal to the strength inside your grains. And above this Iso-strength temperature, your grain boundary presence actually reduce the strength of your material. So that's the reason why that's the reason why in a lot of high temperature applications, you actually prefer a coarse grained structure rather than a smaller grained structure because of the grain boundary softening, okay? So in today's lecture, we have talked about the strengthening mechanism by grain refinement. Thank you very much. [MUSIC]