 In the discussion of mechanical behavioral materials, we will take up a overview of some of the properties, how they arise, the mechanisms behind these properties and further we will try to discuss how these properties change when we go from the bulk to nano materials or nano structured materials. We will also take up issues regarding artifacts in testing artifacts in the properties which we observe and try to describe that how we can develop better strategies of testing materials and also coming up with properties which are specifically peculiar to nano materials. Some of the properties which we will be talking about are elastic properties, plasticity, we will also talk about strengthening mechanisms, we will talk about the switchover from twinning to slip, we will talk about important properties like stupeplasticity and creep in nano materials and also about the role of grain boundaries in nano structured materials. In the whole process, we will also take up some hybrids like nano composites which are lot of research is being conducted in these areas nowadays. So, we will alternate between the basic property in bulk materials, the description of the property and then take up specific properties of nano materials. Now, it is important to start with a broad overview of mechanisms by which a material can fail. Now, this can be mechanisms which are physical, they can be electrochemical or they could have other origins from which failure can be material can fail. In this context, failure has been defined as a change in the desired performance which could involve changes in properties and or shape. As you know, there is a component like a gear wheel which is meshing into a system with other gears in the system, then if there is a change in the shape of the gear then automatically this is constituted as failure. Often elastic deformation is not included in failure, but it is obvious that suppose I have a spring board from which a diver is going to take a dive into a pool, if the spring board deflects too much even though this would come under the class of elastic deformation, needless to say this would be a failure of the desired performance and therefore, it would constitute failure by elastic deformation. The common other methods of failure would consist of plastic deformation in its slip is one of the important mechanisms of plastic deformation, fracture wherein the material cracks propagate in the material leading to failure, fatigue could be one of the reasons behind such a failure and fatigue could also lead to cracking in the material. There is creep and we will see that there are many creep mechanisms which can actually lead to failure. Microstructural changes itself can lead to failure like for instance suppose I can think of twins and stresses related to twins, phase transformations, green growth, particle coarsening which we have taken up in detail already, all these can be microstructural changes which would constitute failure. Needless to say chemical or electrochemical degradation of material which includes corrosion and which includes oxidation etcetera would also can or can also lead to failure in the material and therefore, because once you have corrosion that means that the material is getting degrade on the surface and surface being an important part of the whole material can lead to a desired I mean undesirable change in the properties of the material. Physical degradation like wear and erosion can also lead to failure and this would be very important when you have two components in contact like the gear wheel where in surface wear could be an important property which we need to take into account. Suppose you are talking about blade operating under water which is propelling a ship then erosion from particles in the medium could be a serious issue and therefore, that also need to be protected against. Having got a broad overview of the mechanisms by which a material can be I actually fail. We will take up some of these during the course of this lectures where we will see that how some of these properties have a beneficial effect when we go to the nano scale and therefore, we can actually improve the properties when you work in nano structured materials. Parallel we will also highlight some of the problems in using nano structured materials and therefore, putting together this problems and the advantages we of course, design strategies for improving the performance of materials and components. The first property we take up is a physical property which is density, but we take this up because this is closely related to some of the mechanical properties like the elastic properties. Now, suppose I am taking a bulk material then the fraction of grain boundaries in such a material and the fraction of what you may call triple lines is very small and therefore, we can ignore them in terms of the overall contribution to the elastic properties. Of course, we cannot ignore grain boundaries when we are talking about plastic properties and we will take up those important contributions later, but as far as elastic properties go we can often ignore the grain boundary contribution. The problem with nano structured materials or nano structured bulk material is a challenge because often full density or close to the theoretical density cannot be obtained and residual porosity can affect the density. So, in our arguments we always have to keep in mind that when you are talking about density of nano structured materials there could be an issue of porosity and this porosity could drastically affect the properties and one of them being the elastic properties. And we would like to get rid of this artifact which is coming from the specimen processing or specimen preparation. Now, before we take up the elastic properties of a nano crystal it is important to note that what is a grain boundary and what are the higher order entities in a grain structure. Now, I can visualize that of course, you have a single crystal then you have only one defect in the material which is a surface. Suppose, I have a multiple grain material and I draw it schematically here then I can notice that each one of these is a grain and one grain can be differentiated from another grain. So, this is grain one and this is grain two another one grain can be differentiate from another one by the orientation difference across the grain boundary. Let me highlight the grain boundary by red lines and we have already noted that grain boundaries are regions of lower coordination as compared to the bulk. So, suppose this were FCC copper crystal then the grain would have a coordination number of 12, but the grain boundary would have something less than 12 and therefore, they are regions of extra free volume and lower density. Now, apart from these grain boundaries as you can see there are higher order terms here or higher order structural entities like this is what you might call a triple line. So, in two dimension of course, it looks like a point, but in the third dimension this is a line and therefore, this is a triple line. So, you have grain boundaries and triple lines further you can also a conjecture height that if four grains are meeting at a point then that would constitute a quadruple junction. And in terms of the energy and bonding you can think of the grain interior being of the lowest energy the grain boundary being of a little higher energy the triple lines being of even higher energy. And finally, quadruple junctions being of the highest energy if you take a bulk material then the volume fraction of each one of them is going to be small and it can be ignored, but in nano structured material this grain when I am talking about a nano structured material I am talking about a bulk nano structured material which means that the grain sizes of the order of nanometers. So, this is of for instance maybe a 10 to 100 nanometers and therefore, in such a material the effect of these on the density and elastic properties cannot be ignored. Obviously, needless to say their effect on plastic properties and fracture and other behavior also cannot be ignored. Therefore, let us see how these entities affect my density. The coordination number and packing close to the grain boundaries and which also includes triple lines which I short form called TL quadruple junctions the QJ's and this quadruple junctions sometime in literature is also called corner junctions is and is expected the coordination number and packing is expected to be lower than the bulk in or if you compare it to the single crystal. This implies that on decreasing the grain size the density of the sample will decrease. Though this effect is expected to be a marginal effect this is not going to be a drastic effect, but nevertheless when you decrease the grain size in other words you increase the amount of grain boundaries and triple lines per unit volume then you expect that the density would actually decrease for a polycrystalline nanostructured material. However, this effect that is the reduction in density with grain size is expected to become noticeable when the grain size is reduced in the nano scale regime and this is an important effect which sort of like shows this prominence only when you go to the nano scale regime as we shall see shortly. Now, how to understand the effect of these the way to understand is that we assume a certain grain morphology and then from that grain morphology we can assume that there is a distribution of triple lines quadruple junctions and we can therefore now calculate the density based on the distribution of these. One of the most common ways of understanding grain structure and grain boundaries the way it is modeled is to assume models of the form which I am showing here. In this case you can see that there are two kind of standard packings of grain structures which are used this is using something known as the tetrachydecahedron sometime also referred to as the Kelvin's tetrachydecahedron which is a space filling solid and here I have got something known as the rhombic dodecahedron which is also a space filling solid. Now, you can see that how these rhombic dodecahedron tetrachydecahedron can monohedrally implies using the just single tile a single shape can fill space. Now, the reason for using these shapes is that this is mathematically easier to model that means I have a single shape and I can use a entire crystal to be made of grains of or the polycrystal to be made of grains of this shape. And the difference in choice comes because this solid even though it is a preferred choice for a grain structure some people also prefer to use the rhombic dodecahedron because every phase has identical shape which is a rhombus in this case there are two type of phases one is a hexagonal phase and one is a square phase. And therefore, there are two choices which are classically used for modeling grains and to calculate effective entities from grain boundaries and also the mechanical behavior of grains. But in the calculations as we shall show now we will use a different kind of a shape for a grain and this shape we are going to use is a cube the reason to you for us to use a cube is to make it mathematically simple to actually calculate the various quantities which we are interested in. As a first approximation we shall assume that the grain boundaries have a similar character in nano structured material. So, this is a very drastic assumption, but we will make the assumption that even though we are going to nano structured material the overall crystal structure of the grain the grain boundary structure the grain boundary width etcetera remain very comparable or very similar to that in a micron sized material. So, this is a drastic assumption we are making, but this will help us make a simplification in terms of the calculation. Additionally, we will assume that we will the value of a grain boundary thickness which is now a very what you might call a quantity which is based on a definition the region close to a grain boundary where there is a disturbance of the atoms from the expected lattice position I call it a grain boundary region. And this is typically less than about one nanometer, but for simplicity here we will assume that it is about one nanometer. And in metals typically it has been found that it is of the order of about three atomic layers. So, to proceed further we will make two assumptions number one is that the grains are all cube shaped which is of course, a very drastic assumption because as he said the closest they actually can come to is something like the Kelvin's tetrachydecohedron and definitely not close to a cube. The second assumption we will make is that the character of the grain boundaries the character of the triple lines and quadruple junctions do not change as we go from a micron size material to a nano size material. The third assumption we will make is that the grain boundary thickness is which is the region of disturbance around the grain boundary or a triple line is of the order of one nanometer. Making these assumptions we will go ahead and make a model to calculate the fraction of grain boundaries with decrease in size. Now, we had talked about porosity we will be ignoring porosity in all these calculations, but as we pointed out that in some materials when in so called bulk or fully dense material as reported in literature could actually have about 3 to 5 percent porosity. And this 3 to 5 percent porosity would actually dominate over some of the effects we will be talking regarding grain boundaries in triple lines, but nevertheless we will assume for now it is absolutely theoretically dense except for this grain boundary and triple lines etcetera. And there is no contribution of porosity to the overall effect of reduction in density or therefore, it is influence on elastic properties. Now, therefore, now I have cube shaped grains which is filling space and the region between these cube shaped grains is the grain boundary and additionally we of course, have triple lines which we can plot like this. So, these are triple lines and the region between the grain boundaries is the grain this of course, is the grain boundary region in this model and we are showing a 2 dimensional cross section of the 3 dimensional crystal structure. So, this is my size of the grain and this is my thickness of the grain boundary. Now, you can see that this such a structure can be described by unit cell which consists of this yellow region and the blue region around it. So, this is a unit cell which is styling to give rise to the entire structure. The fraction of grain boundaries of course, and of course, triple lines and quadruple junction depends on the grain morphology. For simplicity we have assumed that a cube morphology is the one to make our calculations very simple. And in cube grains triple lines can also be referred to as triple junctions as often done in literature. The fraction of triple lines and quadruple junctions become important when the thickness for about thickness of about 1 nanometer, when the grain size is of the order of 10 nanometer or below which we shall see very soon from the plots and the way we make the calculation. The fraction of grain boundaries as a function of the size of the grain which we have shown here d and the size of the grain boundary which is t can be written as d cube divided by d plus t the whole cube. This is a pure geometrical calculation based on the cube morphology. Defraction of the grain boundary region can be written as d cube by d plus t the whole cube 1 minus this value. The fraction of quadruple junctions can be given as 6 t cube divided by 8 into d plus t the whole cube. The fraction of triple lines can be given as 6 t square d divided by 4 into d plus t the whole cube. Therefore, I have definite formulae for the fraction of the grains, the grain boundaries, the quadruple junctions and the triple lines. If I plot these functions as a function of the grain size you would notice that this is the main function which is the fraction of the grain. The fraction of the grain seems to reduce below as you go down to smaller and smaller grain size. So, that means that if the fraction of the grain is reducing that means that the remaining of the material is going to increase which is now going to be the fraction of the triple lines, the fraction of the grain boundary and quadruple junction which you can normally see that they are increasing. The important point to note from this graph is that it is only below about 10 or 20 nanometers. So, I can draw some vertical lines around this 10 nanometer or 20 nanometer. So, in this regime which is below this size that grain size effects start to dominate on these other entities of the crystal. That means if I am at a larger grain size and about 15 nanometers and I am in a larger grain size then I can safely ignore the contribution of these two density or as because the volume fraction is going to be small then therefore, I can ignore them. Now, what happens to these quantities when you are in the small grain size regime and we already defined the small grain size regime to be about less than about 20 nanometers. You notice that the grain boundaries already start to play an important role when you are already less than about 30 nanometers. While the importance of triple lines and quadruple junctions become very important only when you are below about 10 nanometer. You can see that these function start to dominate only in this regime. This is very important because when you are talking about nano crystalline materials with the grains of about 10 nanometers which are plastically deformed. You will notice that you cannot anymore ignore the presence of triple lines and quadruple junctions and they will play an important role in the plastic deformation of the material and of course, first before we reach that stage we will of course, going to talk about the elastic deformation of materials with these kind of entities in there. To summarize this slide and to summarize the lesson so far with decreasing grain size the importance of quadruple junctions grain boundaries and triple lines is going to increase. So, the grain boundary can be thought of as a 2 D structure. The triple line can be thought of as a 1 D structure and the quadruple junction can be thought of as a 0 D structure. So, you can see that these 2 D 1 D and 0 D structures actually start to play a prominent role only when the grain size is below about 15 nanometer or more specifically about less than about 30 nanometer. The grain itself is a 3 D entity as we know, but more specifically the role of triple lines and quadruple junction the 1 D and 0 D entities in the structure start to dominate only below about 10 nanometers. And the important point is that we can actually these calculations are coming from assuming a cubomorphology and we are assuming a grain boundary thickness of about 1 nanometer and then we can write down the fraction of these various entities given by these formulae by a pure geometrical calculation which is easy to perform. Therefore, I can have a fraction of these entities and I can actually calculate how these fractions start to change. And the overall trend is that whenever whatever fraction is lost from the grain goes into either the grain boundary or to the triple line or the quadruple junction. Now, what effect does this have on the density of the material because we already noted that the bonding the coordination number are actually small or lesser as compared to the bulk in the grain boundary or the higher order entities. Therefore, now this is going to affect my density and as expected this density is going to be this effect is going to be prominent about 10 nanometers or of that order. Now, I can write down the density of the material using the formulae that rho is equal to 1 minus fraction of grain boundary fraction of triple line fraction of quadruple junction into density of the grain. That means, if I take out all the other entities what I am left to the fraction of the grain grains and that fraction I can multiply by the density of the grains. And then I can multiply the relative fractions of all the other entities like the grain boundary the triple line quadruple junction with respect to their relative densities. And I will get the density of the entire composite of or now the poly crystal. So, nano crystal I will get I will get the density of the nano crystal. Now, to get this number I need to know the values of f g b f t l and f q j I need to sorry I need to know the value of rho g b rho t l and rho q j having I have already calculated the fs the volume fractions already in the previous slide. So, I need to know these entities and this can be for this of course, you need to do some detailed modeling may be some kind of an atomistic modeling or a molecular dynamics modeling. But, from some kind of a reasonable estimate which is arrived at by looking at various models and also experimental results is that it is assumed that the grain boundary density is about 0.95 the density of the grain the that means, now I am talking about relative density of the entity with respect to the grain. So, suppose the relative density implies the density of the poly crystal divided by the density of the grain. So, for the individual entities grain boundary has a 0.95 relative density with respect to the grain the triple line is about 0.90 and the quadruple junction is about 0.81. That means, as you go reduce the dimensionality from grain boundary to quadruple junction. So, this is 2 d this is 1 d and this is 0 d then I see that the density actually decreases. Now, these numbers of course, as you as I just pointed out are not sacrosanct numbers and it depends on the model depends on the material system it depends on the what you define as a the grain boundary region etcetera and how you calculate the density. But, for now we will assume some kind of a numbers to arrive at what you might call a qualitative picture of where in can I see a change in relative density which finally of course, I want to correlate with the reduction in the modulus of the material. And again I want to point out that we are actually ignoring porosity in this whole calculation because if porosity is present that is going to be the dominant factor as compared to some of these effects which we are talking about. Needless to say the fractions of all these put together has to going to be 1 from which we have actually used to calculate this number at the top. Now, if I plot this relative density using these values for the grain boundary triple line and quadruple junction I notice that the relative density actually decreases with grain size and you see that below about 20 nanometers this effect tend to dominate. So, when I am below about 20 nanometers I see that I have to worry about the presence of grain boundaries in triple lines in the calculation of relative density. That means, now the density of a bulk crystal if I is 1 then the density of a 10 nanometer grain size crystal is going to be about or let me take a little smaller number. So, that here is going to be about 98 percent that means point about 2 percent reduction in density has taken place purely from the fact that now I have a poly crystal in which there are going to be a higher density of grains grain boundaries triple lines etcetera. Therefore, now I have a poly crystal what you might call a porosity free or an artifact free poly crystal purely from the internal geometric structure which involves grain boundaries. There is a reduction in the relative density and this relative density reduction can be pretty high if I are going to smaller and smaller sizes. So, now what is the effect of this on the elastic properties we will try to study that and for that we will have to invoke the concept of what is called the elastic properties of a composite. Because now I can think of my poly crystal as a composite between a grain and all the other structural entities we have just now talked about. But before we go to that let us talk about elastic properties of certain other nanostructures before we take up the elastic properties of the poly crystal we just said and there are some very interesting examples of nanostructures and nano composites wherein elastic properties seems to play a very important role. We first start we will assume an isotropic material that means the properties of the material do not change either with position or direction and we know that for an isotropic material we can describe the elastic properties using two independent constants e and nu or g and nu and therefore, it makes our overall understanding pretty easy. If you the introduction of an isotropic properties of course, brings richness to the system, but makes our understanding more difficult therefore, we will stick to isotropic properties for now. For a cubic crystal we know that for instance in if you actually talk about the you need three independent elastic moduli to describe the elastic properties of a cubic crystal and if you go down to lower symmetric crystals then you need more and more elastic moduli or elastic constants to describe the properties of a anisotropic crystal. And this assumption is drastic as we shall see for now because we already noted that if you take a poly crystal a nano poly crystal the actually the density and therefore, the modulus will vary from position to position as you get close to grain boundary expect a different modulus as compared to that in the bulk of the grain. Often modulus of some nano structures are as reported below it should be noted that moduli are bulk macroscopic properties and the definition is extended to be applicable to these structures. So, we should note that modulus as such as a property is a bulk macroscopic property that means it is an average over a large number of atoms or a collection of atoms. And if I am going to be describing modulus of a small entity like a nano structure like a carbon nano tube which could be a single wall or a multi walled carbon nano tube or I am talking about a zinc nano wire where in the number of atoms are much limited. Then I am actually extending the definition of what is called the bulk elastic modulus to some of these smaller entities. But nevertheless having extended this definition let us note that what are the interesting properties we can get by working with these nano structures. Some of the values reported in literature I am showing here below. Silica nano wires have found to have a modulus variation in the order of GPA 2200 GPA, zinc oxide nano wires from 140 to 200 GPA, multi walled carbon nano tubes in the range of 11 to 63 and this obviously depends on number of walls that the modulus is going to change. Single walled nano tubes have modulus which are very high and which approach that of diamond. That means now even though single walled carbon nano tubes can be thought of as folded graphene sheets but their modulus when of course pulled in the direction of the length is can be pretty high and can in fact challenge that of diamond. That means now I have a entity which is extremely strong which is a single walled carbon nano tube and people are women envisaged making ropes out of the single walled nano tube which can actually take lot of load which is extremely which has a very high modulus. That means that I can use it as a reinforcement in composites and we will call such composites nano composites even though we understand that the grain size is not in the nano it is the reinforcement which is in the nano to actually obtain very good properties. Now to understand the modulus of composites we need to understand that how if I mix two phases then how the overall modulus emerges. There are two ways of understanding this and this depends on if the state of stress in the presence of the composite. So, now let me draw some schematics of composites that actually I could have a second phase which is in the form of spheres or I could have a composites in the second phase in the form of long rods and these rods of course could be aligned or they could be in other forms and we can also think of laminates which from the composites. Now the important thing of course is the interface between these first phase and the second phase which I mark in red here. So, I have an interface so it depends if this interface is actually transmitting all the strain which means that it is in a state of iso strain condition which means the strain in the matrix is the same as the strain in the fiber or the other extreme could be an iso stress condition in which case the stress in the matrix is same as the stress in the fiber. In the case of iso stress we can think of the two materials now suppose this is material one and material two which I draw in two different colors. So, I can think of two configurations wherein the matrix and the fiber materials are as if in a parallel configuration another possibility is that the matrix and fiber which is the one of them is are in a series configuration. The parallel configuration is like an iso strain configuration and series configuration is like an iso stress configuration and therefore, using either one of these two configurations I can calculate the modulus of the composite given the individual moduli of the fiber and the matrix. In the case of iso strain conditions that means the strain in the matrix is same as the strain in the fiber which is the strain in the composite I get a formula that E c is equal to E f V f which is V f is the volume fraction of the fiber and V m is the volume fraction of the matrix which is nothing but 1 minus the volume fraction of the fiber. Assuming that there are only two phases in the material there is no porosity etcetera and this resistance in series configuration gives me a resistant the what you call the modulus of the composite this kind of an averaging is called void averaging and we can get the modulus of the composite. The other extreme of a possibility is the iso stress condition which I pointed out which means the stress in the matrix is same as the stress in the fiber which is the stress in the composite and therefore, this is like resistance in parallel configuration which I pointed out and this leads to what a formula like 1 by E c the E c being the modulus of the composite is V f by E f f the subscript refers to the fiber plus V m by E m m being the subscript referring to the matrix and this kind of an averaging is called Roy's averaging. Therefore, I have two extremes of the modulus of the composite with respect to the volume fraction of the material. Suppose I take any given volume fraction f just for the sake I will call this f 1 here this volume fraction is f 1 then I have an upper bound given by the iso strain condition which is nothing but the void averaging and I have a lower bound given by the iso stress condition which is the Roy's averaging. Therefore, my modulus of the composite is going to lie between the upper and lower bound of the composite. The reason that the real composite is not exactly one of the two is that actually the matrix fiber interface could actually partially be slipping could actually not be contiguous in places and therefore, not knowing fully what is the fiber in each and of those fibers and their orientations and their characteristics of the interface. I can only tell that the overall average has to be between the upper bound given by the iso strain condition and the lower bound found with the iso stress condition. And of course, in typical composites you do not add up to 100 percent of the fiber usually fiber concentration is about between 5 percent 25 percent it is a smaller fraction as compared to the overall matrix in of course, exceptions we may go to higher volume fractions. Therefore, for a given fiber fraction f the modulus of various conceivable composites lie between upper bound given by the iso strain condition and a lower bound given by the iso stress condition. Therefore, now if I am making a composite the modulus benefit I get has to be between these two limits and the composite we will be referring to very soon is going to the nano composites wherein we will be using reinforcement which is in the nano scale. Now, one such example is the case of the multi walled carbon nano tube reinforcement of alumina. Of course, it is better as we have seen that you reinforce the system with single walled carbon nano tubes, but it is easier in many cases to actually manufacture multi walled carbon nano tubes and reinforce that and you can see that the grain size itself is in the of the order of microns. So, this is not a nano grain size alumina, but it is a micron grain size alumina and what is nano in this whole nano composite is actually these carbon nano tubes which you see here in high magnification. So, this multi walled carbon nano tubes have been used to reinforce alumina. The Young's modulus of such a composite has been found to be as high as about 570 g p a and the range typically has been found between 200 and 570 g p a and this of course, depends on the nano tube geometry its length its number of layers in the nano multi walled nano tube the quality of the nano tube and the porosity in the alumina, but if you note that the Young's modulus of alumina is actually smaller than you actually can get a considerable benefit in the modulus by adding a nano tube reinforcement to the ceramic. This additionally has been found to also give fracture toughness in increment which of course, we do not deal with here, but it is important to note that when you add nano tubes to alumina there are other benefits which you get in the overall process. So, this example of a carbon nano tube is nothing but an example of making a composite and therefore, enhancing the modulus of the material by making a composite. Of course, the reason for making a composite as you are aware is that one or more of the properties see a beneficial effect with respect to just the individual components. Suppose, I use just alumina or just carbon nano tubes then whatever the properties I get the composite actually gives me a added beneficial property up and above each one of the components and that is why I am justified in using a composite because some of these composites are actually difficult to process and synthesize and actually use in routine industrial production. Now, we are told that a nano poly crystal can be thought of as a composite between grains and grain boundaries and triple lines and other kind of a defects. Now, for a first order we can actually think of a nano poly crystal to be having we are composite of the grains and these other defected regions in the poly crystal which we saw just now that has a lower density as compared to the bulk of the grain. These regions obviously are also expected to have a lower modulus because the bonding is inferior as compared to the bulk of the crystal. Now, we notice that there is a bulk change in modulus only when the grain size is about below about 20 nanometers and we had we continue the modeling which we did before. We assume a cubic morphology and we also assume that the grain boundary has a modulus about 0.7 that of the grain. Here again if to really determine the correct modulus of the grain boundary is a difficult task. There is obviously no such single number which can be thought about because grain boundary itself we have seen is not a single plane, but it is a region in the material and therefore, we have to assume some kind of a number and for now we assume that the e grain boundary is about 0.7 e grain. And as before we ignore porosity in the material because we as we shall see in the next slide porosity could be a serious issue in very many problems and cause very many problems. Early results showed that if there is a reduction in modulus about for about below even about 200 nanometers. That means that the region we have already seen that where grain boundaries and other kind of entities do not play an important role, but they did observe some reduction in modulus when they were already below 200 nanometers. And if you remember the slide from before that all the drastic changes seem to happen only below about 30 nanometers and not below about 200 nanometers. And in fact in the context of bulk modulus of a polycrystal 200 nanometers can even be thought of as a bulk material. You can safely ignore most of the other entities in it the entities implying the grain boundary the triple lines etcetera. These samples were perhaps this result was perhaps because of porosity in the samples and the investigators actually did not characterize a fully dense sample. So, we now can understand that perhaps in hindsight that these were not really bulk sample fully dense samples and that was giving rise to the change in modulus below 200 nanometers. And if you really using the value of e g b is 0.7 e g and I have a relative modulus defined as a e polycrystal by e grain. That means it is the reduction in modulus with respect to that of a grain and I plot the relative modulus with respect to grain size. I see that most of the changes seem to take place when you are around about 20 nanometers or below. So, this seems to be the region where actually these other entities seem to play an important role and there is a reduction in modulus. Since I am assuming a composite I obviously have to have an upper bound and lower bound as shown here. The upper bound corresponding to iso strain and the lower bound corresponding to iso stress and I see that for a 10 nanometer crystal for instance I can draw a dotted line like this. The lower bound is about 0.9 and the upper bound is about 0.92 or 0.93. Therefore, by using my understanding of composites I can actually calculate the modulus of a polycrystalline material and I see that the modulus of a polycrystalline material is actually lower by about 10 percent when you go down to small sizes which is about 10 nanometer or below. But again I have to reemphasize that in such cases actually synthesizing a porosity free material is actually difficult and whenever porosity is present that is actually going to dominate the effect and not these other effects which I have just now described. There is another important clarification which is required because if somebody needs some old literature it there is some effect which is described as super modulus effect which was observed and there was lot of excitement related to that. And this was these observations told that elastic properties as large as more than 100 percent enhancement elastic modulus of multi layers. So, I am talking about thin film multi layers and they saw a huge increment in the modulus and they called this the super modulus effect. Now, later investigators of course could not confirm the super modulus effect and this work showed that this could be because of artifacts and anomalies. These are not perhaps true effects coming from the multi layer model life, but there could be other effects coming. And reasonable assumption perhaps today would be that there is an enhancement perhaps in these multi layers, but that is much smaller than the super modulus effect which was originally described, but more work has to actually clarify that how general is this trend of about increment in modulus is it only related restricted to multi layers or there are other effects which we need to take into account. But clearly that super modulus effect is no longer attributed as a real effect in nano materials and therefore, it can be ignored. The next topic we take up after elastic properties is plastic deformation of crystalline materials. And often slip is considered as the most important plastic deformation mechanism, but we should note that there are other important mechanisms as well and some of these become important like for instance twinning at low temperatures especially in materials like V c c materials where in slip can be limited at low temperatures. Additional mechanisms could be phase transformation and here when I am talking about plastic deformation I implying that there is permanent deformation in the material that means even in the absence of the load there is a deformation which has been caused which is going to be permanent. And therefore, phase transformations can lead to permanent deformation in the shape of a material. There are creep mechanisms like grain boundary sliding vacancy diffusion and dislocation climb and there are other mechanisms as well which can give rise to find permanent or plastic deformation in a material. Initially we will of course, start with slip try to understand the role of dislocation motion in slip. We will understand also why crystals are weak in for instance in a tensile test. And then later on understand how the behavior of nano crystalline materials and nano single crystals are different from that of the bulk materials which we have been discussed. We have to of course, note that in amorphous materials that the plastic deformation mechanisms could be very different and this could involve shear banding or even viscous flow which is like a almost like a Newtonian flow liquid flow. And this typically in for instance bulk metallic glasses it takes place about the glass transition temperature of the bulk metallic glass. So, we have looking here at a broad overview of all the plastic deformation mechanisms in crystalline materials, but we will essentially focus on two or three of them initially on slip later on on little bit on twinning and further down the lectures on creep mechanisms. If I want to understand the plastic deformation by slip the simplest test I can do on a material is the what is known as the uniaxial tension test. And when I perform a uniaxial tension test of course, I get what is called a load stroke data. That means I apply a load and I get basically some kind of an elongation or increase in length of the material. Initially of course, this load stroke data of course, can be plotted as something known as an engineering strain versus stress plot or it can also be plotted as a true stress versus true strain plot. There are two ways of plotting this data and the way it is plotted can make the curves look very different. As you can see this curve on the left goes up the engineering stress strain curve goes up then it finally, starts to come down and finally, there is a fracture while the true stress strain curve always keeps going up and finally, till there is fracture. Now, there are lot of important properties you can actually calculate from this uniaxial tension test which includes for instance the yield strength, the ultimate tensile strength, the fracture strength, the and whenever the word strength is used in this context it implies a fracture stress or ultimate tensile stress. And you can also calculate for instance the elastic elongation, the elongation to the ultimate tensile strength and finally, the elongation beyond which the material fails which is called a elongation for fracture. Now, the important things to note from this curve are initially there is a linear region in this curve which we call the elastic region and there is a corresponding elongation which is now my elastic elongation given by this value E u. Now, the value of E u and epsilon elastic or E elastic and epsilon elastic is very, very similar because in small strains I can actually use the engineering strain as well as the true strain and there are equivalent only when the strains become large that I have to differentiate between engineering strain and true strain. Now, if you want to divide this curve into parts the linear part can be thought of as what is called the elastic region, but we have to note that this elastic region is macroscopically elastic. In other words if I really want to know truly microscopic elastic region, then that is related that is only occurs to a very small strains of the order of about and I mark a point A here and this is the strain of truly elastic this strain is extremely small. And therefore, beyond that the region though the curve appears straight from A to Y. So, the point A to Y the curve is straight. So, this is macroscopically elastic, but at microscopic scale there is actually some plasticity which is already taking place. The reason behind this plasticity we will see soon is motion of dislocations leaving the crystal or the green boundary. Now, if I carry on maniacal tension test beyond this point Y, then I see that the curve goes up in the case of the engineering strain diagram and comes starts to come down. This peak is where we attribute the formation of a neck and this neck is forming because there is localized deformation. That means that deformation is not spread across the entire gauge length which icon can call L 0. So, there is a localized deformation and this localized deformation finally leads to fracture and the material separates into two half. And in a ductile material typically you will observe something known as a cup and cone fracture and of course, in extremely ductile materials you may even observe something known as a rupture where in the specimen thins down to a point and then it fails. Now, this region blue region of the curve though often it is casually called the plastic region we have to note that is actually elastic plus plastic and that it becomes obvious if I unload from any point like a point like n or a point like m here. Where in if I unload this curve then the curve will unload parallel to my Y axis and I will get an elongation. So, let me go down to the board and draw this again it is to avoid cluttering this diagram. So, the elastic deformation I will exaggerate here. So, that it becomes easy to visualize the straight region of the curve this is exaggerated actually the slopes are of the order of the gigapascal. And now if I unload from a point say m in the curve then of course, I can draw a vertical line and this is now my engineering stream true strain sorry engineering true strain true stress diagram. And I will now unload from point p and I can I will notice that the unloading will take place parallel to this line that slope of these two lines will be same. And if I notice then these are two similar triangles, but this implies that this implies that actually this region which is the blue region that means by drawing similar triangles and unloading the unloading curve being parallel to the original elastic line. I observe that actually I recover more strain here as compared to if I unload from point y. So, in some sense in the sense of the strain obtained the so called plastic region is more elastic than even the elastic region. And therefore, I need to call this plastic so called blue part of the curve as actually elastic plus plastic and not nearly plastic. There are few things of course, we need to ask is that why is this curve going up the blue curve. And this phenomena that the stress actually required for flow increases with strain is called work hardening or strain hardening. And that means the material is getting harder and harder as I am plasticly deforming it till of course, you reach a point like fracture. The reason is as we peak ahead is because of motion of dislocations and the increase in dislocation density that and this intersection of dislocations making dislocation motion difficult. Therefore, this is a result of what is known as a uniaxial tension test. And to summarize the important points there is a truly elastic region from origin o to point a. There is a linear portion which we call the macroscopically elastic region, but between point a and y this specimen is actually microscopically plastic also. From y to f the material is undergoing strain hardening which is nicely seen in the case of the true stress, true strain curve where in the curve is constantly going up. There is a point n in the engineering stress strain curve after which the curve starts to stoop down. And this is the point of onset of necking that means the deformation is very local. And we will see that what are later on that the strain hardening exponent is going to play an important role in determining the amount of strain you are going to get after necking. That means the overall ductility is going to be dominated by a factor which is n which we will see later which is called the strain hardening exponent. Now, what are the variables in a plastic deformation? So, here we are sort of like getting a broad overview and a general feel of the plastic deformation of materials the details of which of course can be consulted to a basic course in the area. Stress strain, strain rate and temperature and normally the effect of strain rate comes into play only when you increase strain rate by a few orders of magnitude. That means suppose I am conducting a test at a strain rate of 5 per second and double the strain rate of 10 per second then this is not expected to affect many of the properties including ductility or the ultimate tensile strength. But suppose I increase the strain rate by a few orders of magnitude from 10 to say 10 power 4 then I do expect that the overall behavior of the material is going to change. Temperature is a very important parameter in plastic deformation and when I am drawing curves like these I am assuming that the temperature is constant. That implies that I am and this temperature at which the those stress strain diagrams are seen are typically at room temperature and at elevated temperatures you would notice that the behavior of the material is going to drastically change. Now, there are some standard or there are some standard equations some kind of an empirical equation which can be written connecting some of these variables important in a plastic deformation. At low temperatures you can write and this is one of the standard equation stress sigma which is now true stress can be written as a constant into epsilon which is a strain power n. And this we are assuming is conducted at constant strain rate test at constant temperature k is called the strength coefficient and n is the important parameter which is called the strain or work hardening coefficient or the work hardening exponent. For copper and brass n is about 0.5 that means that this material can be given large plastic strain much more easily as compared to steels which is which has a smaller strain hardening exponent. Now, what is the importance of the strain hardening exponent is that if there is a localize deformation like making then the strain there is larger and that implies because this is n sits in the exponential that means the stress is going to be larger therefore, further plastic deformation. And this implies that the softer region of material adjacent to the region where you actually put more strain which became harder it means to draw schematically here suppose had a material like this with a neck. So, this region has undergone more plastic deformation as compared to this region and this implies if with larger n this material will be harder as compared to this region of the material. That means this if because this region is softer the plastic deformation will tend to spread here rather than get localized here therefore, I will get more tensile elongation if I have a larger n that is why a material like brass with a higher strain hardening exponent like 0.5 gives you more plastic strain in uniaxial tension as compared to steels with a lower strain hardening exponent. At high temperatures now when we are talking about high temperature we are talking about a temperature above the recrystallization temperature of the material I it is not strain which is the important variable, but it is strain rate which becomes the important variables a variable. And therefore, I can write an equation at for high temperature deformation as sigma is equal to c epsilon dot power m and of course, when I do these tests I keep the strain and temperature constant for two specimens which have undergone test to have comparable values c is a constant and m is another important index which is called the strain rate sensitivity. This is very very similar to m for low temperature test. So, in low temperature the important variable is strain and the important exponent is a strain hardening exponent at high temperature it is the strain rate which is the important variable and it is the strain rate sensitivity which is the important exponent. Now, if m has a large value then you would expect a behavior at high temperature which is similar to having n having a higher value in the case of low temperature deformation. If m is equal to 0 that implies the stress is independent of strain rate and the curve would be same for all strain rates. If m is higher and typically for metals it is a it is a value of about 0.2 then you expect that the material would strain rate hard and more compared to material with lower strain rates sensitivity. If m is in the range of about 0.4 to 0.9 the material may even exhibit super plastic behavior. We will of course, see what is super plasticity and under what condition we will obtain super plasticity later, but essentially super plastic behavior implies that you are getting very large elongations of the even of the order of 400 percent or 500 percent or 1000 percent which is not seen for the case of a normal material like steel or even for aluminum. That means strain rate sensitivity is going to be an important parameter for me in controlling giving me the dictating the amount of elongation I am getting at higher temperatures. In if m is equal to 1 which is the extreme case the material behaves like a viscous liquid or a Newtonian liquid which just flows. Therefore, if the effect of strain rate sensitivity is very very similar to that of strain hardening exponent which is at low temperatures. To understand this effect of strain rate sensitivity we will dwell a little deeper now. We have already seen that stress can be written as C epsilon dot power m and the test being conducted at constant train and temperature. Now, in some materials due to structural condition and high temperature, necking is prevented by strain rate hardening. So, the effect as I told you about n at low temperature is the effect that it actually uniformly spreads your deformation and therefore, localized necking is reduced. Similar role is played by m at high temperature and we will try to understand that how it happens. So, we can of course, write C epsilon dot power m and load per unit area which is the definition of stress of course, and I can rewrite the same equation of using this equation I can put sigma is equal to C epsilon dot power m. And therefore, I can write epsilon dot power m is p by c into 1 power 1 by m and 1 by power 1 by m and from the definition of true strain rate which is epsilon dot is sorry the m should be cut out. So, epsilon dot is 1 by l d l by d t and therefore, putting these two equations together I can get a relationship of d a by d t which is nothing, but the decrease in the cross sectional area per unit time. So, that because the negative sign this is a decrease. So, it is a decrease in cross sectional area per unit time that means, how fast is the necking proceeding. If of course, if this overall specimen remains without necking that means, that the decrease is going to be small, but because there is a necking taking place in a selected region this implies that there is going to be a increased reduction in area in the localized region which we call neck. Of course, d a by d t depends coming from these two formulae putting these two formulae together and arriving at this derivation is p by c power 1 by m which are all constants of course, and 1 by a power 1 minus m by m. If m is small smaller the cross sectional area the more rapidly the area is reduced. That means, if I have a small strain rate sensitivity then the smaller cross sectional areas will decrease even more rapidly and that is not good a condition for obtaining long elongations. Of course, if m equal to 1 we see that it becomes like a Newtonian viscous liquid, but between the value of small m to large m we will try to understand how this relationship can be understood better using plots. So, I have seen from previous relations of d a by d t can actually be related to a with the power 1 minus m by m. So, if I plot d a by d t which is the left hand side of this equation which is a reduction in area per unit time versus a and try to compare different materials with different m values and these are of course, actual materials will have will be close to one of these curves. If m is 1 by 4 0.25 and we said that this is something in the range where typical metals lie which is about 0.25. We see that for a given area the reduction in area is large on the other hand if I had m of the higher value. That means, the increasing m curves are going from here to here to here along this dotted line as you can see. That means, for a given value of a say for some value of a 1 for instance and if I have a higher material with higher strain hardening strain rate sensitivity then the reduction area would be smaller. So, will actually be on the lower curve rather than the higher curve and if it is even higher like 3 by 4 then you will be on the even lower curve. This implies if I am going to compare different materials with different strain rate sensitivities it is better for me to have a higher strain rate sensitivity because in that case the reduction in area per that means, d a by d t which is reduction area per unit time would be smaller. That means, that now my deformation is not going to be localized and therefore, the deformation is going to spread across the sample and therefore, I can get higher elongations. Now, if you look at experimental results done on various specimens you would notice that the curve on the left which I am drawn in the box here. You can see that the percentage elongation versus m you see that if m is larger and now this is now comparing various materials under various conditions there you can arrive at a schematic trend line that if m is larger then you get larger percentage elongation. So, to summarize these few slides plastic deformation means permanent deformation slip and training are two important mechanisms of that. In normal deformation like pulling of aluminum rod under tension it is slip which is the dominant form of deformation. The simplest test actually we can do to understand the mechanical behavior of a material is the uniaxial tension test which is typically plotted as a engineering strain engineering stress diagram or a true stress true strain diagram. We can get important parameters regarding the mechanical behavior like elongation to fracture, ultimate tensile strength etcetera from these curves. The curve can be understood as macroscopically elastic and elastic plus plastic regions. The important variables in plastic deformation are stress, strain, strain rate and temperature. At low temperatures it is strain which is the important variable at high temperatures it is strain rate which is the important variable that means that I do not have to worry about the net strain the material does not accumulate strain in the material is constantly replenishing its microstructure such that at high temperatures you can you do not have to worry about the strain in terms of hardening it is a strain rate which is going to give you hardening. Therefore, there are two empirical formulae which commonly used in literature which is gives you the connection between these variables which is sigma is equal to k epsilon power n where n is the work hardening or the strain hardening exponent. At high temperature you can write down a c epsilon dot power m which m is the strain rate sensitivity and if I want to study the effect of strain rate sensitivity I can actually talk in terms of reduction in cross sectional area per unit time and if I look at reduction in cross sectional area per unit time one of I can I would have a curve for each one of the materials I am studying and for a given cross sectional area the reduction would be more for a material with low m as compared to a material with high m. Therefore, higher the m the smaller would be the reduction in cross sectional area with time. Now, this also can be understood by plotting experimental results for various materials and you see that with increasing m actually you are seeing that there is an enhanced ductility which is obtained. And the explanation is very very similar to what I told you for n for small numbers n for small low temperatures can be now thought of as m for high temperatures. That means if necking is going to take place there is a reduction area there is a d a by d t term here and if the since there is a reduction here the d a by d t here is going to be larger than the d a by d t here where there is smaller reduction in area. And this implies this region is going to get harder with respect to this region if m is has a high value which also implies that this region necking would now spread to other regions or the deformation would spread to other regions and will not be localized to a region where the original reduction in area took place. That means now I will get longer elongation and therefore I can understand the schematic trend line which is been drawn here for these materials.