[MUSIC] Dear students, today we are going to talk about the elastic properties of dislocations. So, up to now, you all ready know dislocations will generate lattice distortions. And this will cause a stress and associated strain fields around the dislocation. And here we will going to talk about the stress and strain fields for the dislocations. And two important comments are going to be made first. So the total stress on the element with a crystal is usually a superposition of all the stresses that it can fill. So both the internal stress and the external stresses applied by external loads. And for an isotropic material, the stress field of a mixed dislocation is then a superposition of the stress fields from his screw and edge components. So here we review some of the basic concepts and the mechanical properties. And, if we have an object and we load it into a uniaxial tensile test or compression test, then we may have this so called normal tensile strain, nominal lateral strain and Poisson's ratio. And object can be also applied by a shear stress resulting in a shear deformation as characterized by this displacement and this angle. And the shear strain is then defined by the shear displacement divided by this dimension. And here, just for the calculation of the stress and strain fields of dislocations, we do some mathematical trick by rotating this sheared object, so that you have two angles here. And the summation of these two angles is this theta value. So here we define the total shear strain as the two gamma which is the tan theta value. And here is some mathematical background to calculate the stress and strain fields. So this is not required in the lecture but it's important and is helpful for your understanding. So, suppose we have initial position of P, and after the strain, the P position is strained to the P prime position, and the P prime vector is characterized by u. And u is having indices of ux, uy, and uz. And then we will be able to calculate the strain fields of this vector. And in particular, the normal strain in x direction is just the partial ux over partial x. And similarly, the normal strain in y direction is partial uy over partial y and the z direction normal strain is just a partial uz over partial z. And correspondingly, you can also calculate the strain in the shear directions. And as we have learned previously, when you have the strain values, normal strain shear strain. And you can correlate the strain values with the six components of stresses, three normal stresses and three shear stresses by the 6 by 6 matrices of this parameters. And for simplicity, when the crystal symmetry is increased, an independent parameter or independent number of parameters in this 36 parameters they will also decrease. And in the extreme case of isotropic materials, you only have two independent C parameters in the 36 parameter matrix. And you have a correlation between your Young's modulus, shear modulus, and Poisson's ratio correlated by this equation. Anyhow, with the equation in the previous page, we are able to calculate the stress field from the strain field as we've just shown. So essentially, we can calculate the normal stresses and shear stresses with the strain values when we have a vector of strength as shown in the previous page. So, this is the basics. Now we move on to real dislocations, we start from a pure screw dislocation. So pure screw dislocation. So suppose we have an element with a screw dislocation, we only consider the shell here from a radius of r0 to a radius of capital R. The reason that we do this, is because very close to the dislocation center or is called dislocation core, you have a very complex stress and strain fields, because of the lattice distortion. And you cannot describe the stress and strain fields with this linear elasticity theory. So anyhow, we only consider the shell here from r0 to capital R. So when you have the screw dislocation, you know that when you rotate around the z direction by 2pi degrees or 360 degrees, you will have one Burgers vector offset on the surface of your element. So in other words, the displacement in the z direction will be the Burghers vector magnitude times the theta value divided by 2pi, right? For an arbitrary cedar value, the z displacement will be described by this equation, okay? And correspondingly, because it's a pure screw dislocation, you have no displacement in the x and y directions, so ux and uy are 0, okay? And then from the equations correlating your displacement with the strain fields, you can then calculate the normal strains in the x, y, z directions and the shear strains in the different directions, right? And we know from here that the normal strains are all 0, and there's no shear strains in xy plane. And you only have shear strains in certain directions. And this is better shown in this r, theta, z polar coordinate, right? Recall from your college mathematics class, right? The polar coordinate says you have a z direction here and you have a theta and r direction here. However, the only difference between this polar coordinate with your usual x, y, z coordinates is that we have different positions in the polar coordinate, the theta our axis, they're rotating as well. So that, when you use the polar coordinate, the expressions become much, much more simple. And you only have shear strain component in the theta z, direction which can be expressed by this. And you will have a corresponding shear stress in the theta z direction as expressed by this one. And all other shear stress and normal stress values are 0. So several comments on the stress field of a screw dislocation. And the stress field around a pure screw dislocation, as we have shown, only has shear component. So there's no net expansion or contraction of the crystal's volume. And secondly, because there's a pure screw dislocation and you will have this axial symmetric or axial symmetry on your stress and strain fields relative to your z-axis. And thirdly, as we have discussed, these discussions and these mathematical expressions are only valid for the regions outside this dislocation core, okay? So, inside this dislocation core, which is on the order of Burgers vector to four Burgers vector magnitudes within that size. You have a highly, highly distorted lattice and you have to use non-linear, atomistic models in order to study the stress and strain fields in the dislocation core. And the situation for the edge dislocation is more complex because you don't really have this z-axis symmetry, axial symmetry. So without going through the details that just to show the results on the stress fields for pure edge dislocation. And here are also a few comments and different from the pure screw dislocation case you have both normal and shears components. And you will have compressive stress in the regions with an extra half plane and you have tensile stress below the half plane. So that's a stress fields, or in a normal case in the normal directions. And then you will have zero normal stresses on the shear plane while you have the maximum shear stress in the slip plane, okay? And the stress filled in the dislocation core cannot be similar to the screw dislocation case cannot be described by the above equations. So, you can even compile a diagram or a figure like this for pure edge dislocations. So suppose you have one positive pure edge dislocation in the center, and then depending on the relative positions around the center, pure positive edge dislocation. And you will have different signs or different directions in your stress fields. For example, in this region, you will have compression in this directions. And, similar case is you always have compressions in this direction. However, when you below this positive edge dislocation, you will have tensile stresses to the left and to the right directions, okay? So this kind of diagram is very helpful. So depending on the positions you are considering, and you can easily tell from the equations that the size and magnitude of your stress fields, okay? Now we move on to next important concepts, the strain energy of the dislocation. You know that the dislocation will cause local stress and strain field and this will give you a strain energy cost. An easy example is here. So, suppose you have a rod like sample and is loaded by force F and this rod like sample has a cross sectional area of A. And then the energy increase as associated with the force is essentially, F times the small change of your sample lengths, right? And because you are in this linear elastic regime, then the stress and strain curve will be something like this, right? It's linearly related with each other. And F is equal to your stress times your cross-sectional area and then the strain if you take the total length out, it will be this one, okay? So in other words, the work done or the increase in the strain energy of the system is the volume of your sample times your stress with a small amount of difference in your strain. So, if you do an integration on the two sides, right? What you are doing is essentially calculating the area underneath the stress and strain curve. So this is for your one dimensional illustration. For the most conventional three-dimensional case, to calculate the total energy associated with dislocations, you just do an integration of all your stress and strain components and multiplied by your volume divided by 1 over 2, right? It's a triangle, so the area is 1 over 2, stress and strain product, okay? So, by doing this and because we already know the stress and strain fields for pure screw and pure edge dislocations previously. We can then calculate the strain energy density or the strain energy per unit volume of a pure screw dislocation and a pure edge dislocation. So, despite there are different parameters there, this as an important concept. The strain energy density, they are always proportional to Gb square, which is the shear modulus times the Burgers vector magnitude squared, okay? So, several comments, again. So, the dimension R is the size of the crystal when you do the integration and r0 is the size of the dislocation core within which you cannot describe the stress and strain fields with linear elasticity theory. And usually the core energy of the dislocation is much smaller than those calculated by the equations in the previous page. So usually they don't really need to care about the core energy of dislocations. And in crystals containing many dislocations, the effective R is then the distance between neighboring dislocations, okay? And an important concept is this. Previously we already known that dislocations they can react with each other and form new dislocations. For example, in this case you have b1 dislocation reacting with this b2 dislocation and then resulting in a single b3 dislocation. And because they are vectors, the Burgers vectors are vectors. So they have to follow this vectorial manipulation, and this is just a one condition, right? So this is vector calculation. And because you have this energy of this dislocation is proportional to the shear modulus times the Burgers vector magnitude squared, then you should have this energy principle. So that your resulting b3 dislocation is having smaller or lower energy than the total energy of your b1 and b2 dislocations. So you have a net reduction in the energy of your system. So that this reaction is then energetically favorable and is thermodynamically favorable, okay? So essentially for dislocation reaction to take place, you need to fulfill this energy criteria as well, okay? So this is called a Frank's rule. Today we have discussed the elastic properties of dislocations. Thank you very much, bye. [MUSIC]
