 Let us now proceed to the next important component of the microstructure which is residual stress and its distribution. We will see how residual stress deserves a place equivalent to phases and its distribution when it comes to the description of a microstructure and how it can influence the properties of a material. The origin of residual stresses if you look at the slide at the top right hand side can be arising from various physical properties like thermal origin, the magnetic origin could be ferroelectric, ferromagnetic etcetera. It could also arise from geometrical entities, examples of some of which we already in seen and may be some more we will consider very soon. Finally, residual stress has been classified based on the scale at which it exists and typically it has been classified as the macro scale and the micro scale. So, let us start with what is meant by residual stress before we go on to consider the classification. So, in this way residual stresses are those which arise in a body in the absence of external loads or constraints. That is if I of course, put a load on a body or I give a displacement to a body I do expect residual stresses to come about, but the stresses which remain after all the external constraints and loads have been removed is called residual stresses. Hence, residual stresses are those which arise in a body in the absence of external loads or constraints. Now, why is that we need to classify residual stresses? We need to understand that there are multiple origins of residual stresses some of which as we may say can be at the very small nano meter length scales, some of them can be at the length scale of the entire component. Some of them are related to the process which are occurring in the material like phase transformations and reactions while others of them are related to the differential thermal cooling with these sample fields. So, residual stresses can be thought of as those arising from defects and these defects include dislocations, vacancies, interstitials, etcetera. We have included voids and cracks in this classification and we will soon see that what you may say there is a slight difference in the way voids and cracks give rise to residual stresses as compared to dislocations and vacancies and we will talk about that very soon. Phase transformations and reactions can also give rise to residual stresses and we will see that how this kind of a residual stress is important from the perspective of the microstructure and the microstructure evolution. The usual origin of residual stress which we often talk about when we are talking about war page of components, etcetera is what is we call the thermal origin which comes from the mismatch in coefficient of thermal expansion like we have a bimetallic strip and we may heat this strip the coefficient of expansion on either side of this bimetallic strip will be different and therefore, the strip can get to a bent configuration along with the presence of residual stresses. So, I have suppose a bimetallic strip and the coefficient of expansion of one of these is alpha 1 and the other one is alpha 2 and alpha 1 can be greater than alpha 2. Now suppose this strip is heated then this will tend to get into a bent configuration and often even after bending the complete stresses may not be relieved especially if alpha 1 and alpha 2 are not in this planar geometry, but in certain more complicated geometry and therefore, you may have residual stresses coming purely from what you may call mismatch in coefficient of thermal expansion. But an important point to be kept in view when we are talking about residual stresses is the fact that often in an engineering sense it is conceived or it is perceived that residual stress is a bad for the body it can cause war page of the component it could cause failure by propagation of cracks etcetera, etcetera. But we should clearly bear in mind that residual stresses can be beneficial or can also be detrimental. So, both these possibilities exist and we will explore this in a little bit of detail in this coming slides. So, let me summarize this slide number 1 residual stresses are important and they have to be considered within a functional framework of definition of a microstructure. Number 2 residual stresses can be classified based on their occurrence on scale which can be macro scale or micro scale and all the other length scales which pervade between the scale of the vacancy or the atom to the scale of the entire component. They can arise because of various physical properties like on a during a magnetic phase and you have a switching on of a magnetic field. They can arise from geometrical entities and when I mean geometrical entities I mean precipitates etcetera and all these origins have to be kept in mind because the kind of origin will tell us under what circumstances this residual stress is arising and how if I want can I engineer this residual stress. Some sort of a classification of these residual stresses can be based on those arising from defects in the material like vacancies dislocations etcetera. Those which arise from phase transformations and reactions and phase transformation I mean it could include austenite to martensite phase transformation a precipitation reaction or many of the other possible reactions including oxidation. They could be of thermal origin which basically means that you have a mismatch in coefficient of thermal expansion. The most important point when with regard to residual stresses is that they could be extremely detrimental for the body as we shall see using examples but additionally they could also be beneficial to the body. So, for whenever I have a beneficial effect of residual stress I can actually engineer the residual stress. So, that my properties of my component or my material improves. Now, there are two examples given here one is that arising from a dislocation which is the shown in the circular body and semi circular body in the left hand side and the other one is that arising during a due to a phase transformation. If a dislocation is present in a material then there are compressive and tensile stress fields which are present associated with the dislocation and typically these would on the whole cancel out. In other words if I perform an integral like integral of sigma i j d v then this integral over the entire volume will give rise to a 0 net residual stresses. In other words if there is a free standing body and I have regions of compressive stresses then there has to be tensile stresses somewhere else in the body. So, that the overall integral of sigma i j d v turns out to be 0. In the case of a dislocation you can see that the extra half plane suppose it is present it is an edge dislocation which I am talking about here which means that the extra half plane could be present in the bottom side. And therefore, there are compressive residual stresses in the region of extra half plane and there are tensile residual stresses in the region other side of the half plane. Now, this is of course, an edge dislocation and the stress contour we are plotting has here is the sigma s stress contour. The close to the core of the dislocation you can see that the stresses are very high in magnitude they can go up to the order of 0.68 gigapascal and more. On the other hand if I go far away from the dislocation core I see that the stresses decrease. So, I am away from if I go far away from the dislocation core then the stresses decrease and after some time I can think of as these the effective region of the dislocation dies down and I do not have to any more talk about the stress field of a dislocation. This stress field plotted here is that for a dislocation in a finite cylindrical body and of course, half the cylinder or half the circular shown here for convenience because the other half is exactly symmetrical to this half. So, this is a mirror plane here with respect to the stress fields. In an infinite body the entire half space for instance the top half says would be completely tensile and the bottom half space would be completely compressive. But in a finite body you will notice that here there is tensile stresses which means there is a positive value, but farther away from the core of the dislocation actually it becomes compressive. So, this is one nice example which is the stress of the dislocation which is present in the absence of any external loading. Of course, if I apply a external shear stress this dislocation may be driven because of the interaction of the externally applied shear with this dislocation stress field and the figure on the right talks about the residual stresses coming due to phase transformation. Now, of course, depending on the kind of dislocation we are considering a pure edge pure screw or it could be a mixed dislocation the stress field which we are saying could vary considerably. But here we are considered an edge dislocation in a cylindrical body for convenience and if you notice that the size of the body is also very constrained it is actually of the about 50 or 16 nanometers. Now, this defect this stress field we are saying is because of a crystallographic defect which is a dislocation. On the other hand the other example we are considering here is due to a phase transformation. That means that there is originally a parent phase in which there is a small region of material which is transforming to a different phase. For instance in this example we have shown the stress plot for a coherent gamma precipitate forming from a copper aluminum alloy. So, you have a copper aluminum alloy from which a nearly pure gamma precipitate comes out and the stress plot is showing that this gamma precipitate is associated with this what you might call a coherency stresses and therefore, the body is stressed. So, let us consider briefly how what is the origin of these stresses in the board. So, we have a body in which I can think of as a small region which is transforming of course, it need not be a spherical region, but for convenience we will consider a spherical region. Now, suppose this body this volume this or whatever I marked in shaded is actually going to transform from one phase to another. For simplicity I can talk think of it as going for a for instance from phase A to phase B. There is no reason to believe that phase B and phase A have the same volume. Therefore, I can make what is known as a shall be cut. I can take out this region of material and therefore, I will have a circle outside and in this material I will actually have a hole like this. So, this is a hole in a medium. Now, I allow this region of material to transform and the transform volume could be larger smaller. Of course, in general it could be associated with both shear and volumetric strain. So, the new body could have a new shape could also or it could have a new volume bigger volume or a combination of both. In other words shear and both dilatational strains will be may be involved with this phase transformation for now I will just for simplicity I will consider only the dilatational strains. So, the body which was originally occupying a volume B here could be occupying a larger volume or a smaller volume and therefore, after phase transformation say for now it occupies a larger volume. So, this is now the phase A which has got transformed into phase B, but this is now a free standing body which is allowed to freely expand, but in reality this is actually embedded in this other material which is now my matrix material. In other words this volume B this material B has to again fit into this hole from which it arose originally. This implies that the surrounding matrix will actually tries to compress down this volume because this now occupies a larger volume as the example I have considered. And the material we want to expand outward and this will come to an equilibrium configuration wherein a certain for instance I will just draw by dotted line the equilibrium configuration. After the outer body is trying to push it inward the inner expanded body wants to go outward and finally, we have an equilibrium configuration. This leads to a residual stress state in the material which is what I am showing you as the plot. So, let us return to the slides and see how does a stress plot look if such a material were produced by a phase transformation. So, this is what we call a coherent precipitate wherein there is a continuous lattice plane matching between the precipitate and the matrix. Now you can see that the region where the precipitate sits has been stressed in a tensile fashion which is shown by the red color contours. But importantly also that the matrix also has been stressed and here we are plotting actually the sigma y contours here or which is we want to write fully is called the sigma y y contours. And you see that regions of the matrix have been stressed in a tensile fashion and there are other regions in the matrix which are in compressive stresses. But the important thing to note again is that the entire body there is a stress field which that means, not only the precipitate is stressed, but also the matrix is stressed. And this effect of the stress field of the precipitate can have considerable influence the way the precipitate actually interacts with other defects in the material like which also could be an edge dislocation sitting in the matrix. And in fact, it could actually attract a dislocation sitting in the matrix and it may on further growth the precipitate may go from a semi coherent state to a semi coherent state. Where in this edge dislocation from the matrix or a dislocation loop from the matrix could actually play the role of an interfacial misfit dislocation. So, we are clearly seeing that there are crystallographic defects like dislocations which have residual stresses. And unless I remove a dislocation from the material these the dislocation will continue to have these residual stresses. We saw that entities like precipitates can have residual stresses. And residual stresses of one origin can actually interact with residual stress of other origin. Therefore, giving rise to certain interaction of defects and these are long range stress field interactions of defects. One nice example perhaps which we have about which we have talked earlier was the fact that how dislocations which are present in the matrix in a random fashion of the same sign. That is dislocation of the same sign present randomly in the matrix can actually attract each other and form a low angle green boundary when you actually heat the material and this process called recovery. So, I will just draw that figure once again to remind you that how suppose I had a bent crystal in which there are dislocations sitting randomly in the crystal. So, if you do a recovery heating process then these this could come together and arrange in a form of a low angle green boundary. So, there will be a slight misorientation in the angle between the two sides of the plane and this is now an array of dislocation forming a low angle green boundary. So, we had considered this example before. Therefore, this is possible through the of course, through the thermal activation, but also equally importantly by the interaction or the long range stress fields of the dislocations. Now, we go to certain important points regarding what is the origin of these residual stresses and important question that why am I including residual stresses and its distribution as a part of the definition of a microstructure. We have to note that processing will influence all three forms or all three origins of residual stresses. When I am talking about three origins, I am talking about those arising from defects, those arising from phase transformations and those having a thermal origin. Therefore, when I am making doing a thermo mechanical treatment which could be rolling, forging, hot rolling etcetera, which is giving rise to my final product or a component, then this processing will actually affect the all three origins of the residual stresses. So, this has to be kept in mind. Additionally, we also have to keep in mind that these origins also represent multiple length scales. We had declassified earlier we had noted that residual stresses can come from the entities of the micro scale or entity at the macro scale, but these are what you might call very crude classifications. We had noted that across all length scales you could have residual stresses and the length scales we are talking about is right from that at which a vacancy exists to the scale of precipitates and dislocations to the scale of the entire component. So, we have to remember that therefore, residual stresses can pervade across the entire gamete of length scale available to the material. So, we have to note some important points in this regard. The stress fields of associated with GP zones and aluminum copper alloy for example, is the same length scale as a dislocation stress fields. Now, GP zones as we know are copper rich zones in an aluminum copper alloy, which comes about when you actually age a super saturated solid solution. So, these are copper rich zones sitting on specific kind of crystallographic planes, but since copper rich zones have a different lattice parameter as compared to the matrix aluminum copper alloy, which is basically aluminum rich alloy. Therefore, there are going to be residual stresses in the alloy, but this scale of residual stresses or the effective length scale at which these residual stresses reside is of the same order of the length scale as a dislocation stress fields. Now, if you had large cracks in a material, this can give rise to macroscopic stress fields while micro crafts may have a much smaller effective region of stress fields. Therefore, we have to note that an important point with regard to cracks and voids being associated the residual stresses is that actually cracks and voids themselves are not stores of residual stresses. Cracks and voids are nothing, but stress concentrators or stress amplifiers. In other words, if I apply a far field mean stress then a crack present in a material will actually amplify these stresses. And as we know, if there is any other origin of residual stress in the material, for instance as we saw the precipitate and if there are cracks present in the vicinity of this precipitate, this crack will may tend to amplify the present residual stresses. In other words, though cracks and voids themselves are not associated with residual stresses, since a small amount of mean field stress can actually be amplified by a crack, we need to talk about we may want to include cracks and voids another kind of what you might call pores in the material into that definition or into the origins of residual stresses. So, this is very very important. So, once again that cracks and voids themselves are not associated with residual stresses, but as they actually tend to amplify far field or mean field stresses, we have to we tend to include them in the definition of residual stresses or the origins of residual stress. In micron size components, the scale of thermal residual stress is expected to be smaller than that in their large square scale counterparts. So, as we know now that because of nano technology, we have components which are actually in a nano scale, we have components in the micro scale. And of course, we have our usual regular components and devices which function in the macro scale like we could be talking about a gas turbine engine or the blades of a gas turbine engine. These are of course, in the large meter length scale, but there could be other components which are called NEMS and MEMS. For instance, the micro electromechanical systems which exist in a much smaller length scales. And therefore, here residual stresses, the length scale of the residual stresses is much smaller, but that does not mean that the magnitude or the effect of residual stresses could be neglected in these. In fact, if the length scale is smaller, sometimes the effect is actually more amplified and we may actually want to take into account the residual stress in a more rigorous fashion. In the example of GP zones and aluminum corporaloids, the stress strain fields are intricately associated with the distribution of GP zones, that is the phases which further highlights the importance of adopting a definition of microstructure, wherein we are not only talking about the phases and the distribution, the defect structure and the distribution, but also the residual stress and the distribution. So, to repeat the last point, in aluminum corporaloids, wherever there is a formation of GP zones or the theta double pi imprecipitate or the theta pi imprecipitate, the distribution of phases is very intricately associated with the distribution of residual stresses, because this is coming from the phase transformation. And therefore, in some sense the distribution of residual stress is closely associated with the distribution of the phases themselves. And therefore, this definition of residual stress which we adopt or the definition of microstructure which we adopted makes a lot of sense. We will consider a few more examples to see that why we need to elevate residual stress to the level or to be an integral part of the definition of a microstructure. And in these two examples, one I consider the growth and nucleation and growth of epitaxial islands and the second I consider is the presence of precipitate phases in a material. In the growth of epitaxial islands, so what happens is that you have an epit substrate on which there is a small island which is nucleating which grows into the form of these mounds. So, let me draw that using a figure here. So, typically there is a substrate and on this substrate I deposit some material and I could be using a technique like molecular beam epitaxy. And this deposited material has three modes of growth possible. Either it can grow in the form of a layer by layer growth or it can grow first as a layer followed by the formation of an island on top of the layer or the third possibility which we have shown here is a direct nucleation of an island and the growth of an island. Now, initially a small island once I will just focus on one island a small island may be nucleated. And later on with progress of more and more additional more and more material this epitaxial island will grow. And I am talking about epitaxial island I mean that the substrate and the island have a very similar lattice parameter typically which means there is a continuity of lattice planes from the substrate to the epitaxial island. In many of these process what is done is that we do not have only one layer of islands after growing a layer of islands as shown in the bottom most figure here. So, let me consider this figure one for instance. So, here we have epitaxial island which have been grown and after growing this epitaxial islands we may go and put actually what is known as a capping layer on these islands. So, in other words you have a substrate you have an epitaxial island and a capping layer which completely buries my islands into the capping layer. In other words a person looking from the top will not be able to see these buried island layers or the layer containing these buried islands. Now, if I go further and actually try to grow a second layer that means now I can continue my MBE process after I have put a capping layer. I can do my MBE process and try to deposit a second layer of these islands in of course, what would one expect that these islands can randomly nucleate at some other position and start growing. But, usually it is seen that that does not take place and there are preferred directions along which there is an alignment and this occurs because of easy direction of strain propagation and for now I will consider that this is my easy direction of strain propagation. And therefore, what happens is that the next layer of islands nucleates exactly on top of the first layer. So, what is responsible for this alignment of the second layer of islands on top of the first layer this is the presence of these epitaxial residual stresses. So, let me go through the process again I have a substrate on which I grow epitaxial certain islands of course, will islands grow or will a continuous film form is depend on other other issues which we are not considering here. But, once an island has grown on one layer if I put a capping layer and then I go on to grow a second layer of islands. I notice that the second layer will align itself in a particular fashion with respect to the bottom layer and now for now I consider it right above the buried layer. In other words it is the second layer is not randomly growing above the first layer it is growing along some preferred directions. That means, there is something which is preferentially nucleating this second layer of islands what is that it is actually the strain or the strain coming from the residual stresses which is the coherency stresses or the epitaxial stresses. This implies suppose I am talking about nucleation we know that from our classical understanding of nucleation that nucleation is preferred at any of the high energy sites in a material. This could be a green boundary or this could be a dislocation and or this could be a preexisting precipitate. And all these forms of nucleation is called heterogeneous nucleation. And surfaces as I cited surfaces internal interfaces like grain boundaries, interface boundaries, stacking faults etcetera are preferred sites. And these preferred sites could also include crystallographic defects like dislocations and other kind of defects we just talked about like cracks and voids. From this example of heterogeneous nucleation we can see that strained regions in a material can also act like heterogeneous nucleation sites that implies that they have the role which is very similar to that of a defect or of a phase present in a material. Like we see that we could have a second phase which is could or the interface of a second phase which could act like a nucleating phase or a dislocation could act like a heterogeneous nucleating site. Here strain or which is originating from epitaxial stresses is the origin of the what you may call heterogeneous nucleation. And therefore, I would want to consider stress or residual stresses part of the definition of a microstructure like defects and phases and phase distribution. Now, another example for and coming this is coming from the what you may call corrosion and the formation of galvanic cell. In corrosion the formation of a galvanic cell lead to the corrosion of an anode that means an anode preferentially dissolves when you actually have an electrochemical cell or a galvanic cell forming. At the level of the individual phases one or more phase may be an anode with respect to the other thus forming a micro galvanic cell. So, when I am talking about a galvanic cell I could actually have a macroscopic electrolyte system where in there is one electrode which is the which performs a role of an anode another electrode which performs a role of a cathode. But, this is what you may call a macroscopic galvanic cell. But, a galvanic cell could also form at the microstructural level at the level of the individual phases. For instance in the diagram shown in the right you can see that there are two phases one has been shaded dark which is now I can call a phase A and there is a fade which is a lighter shading which is a phase B. I assume that this is present in a blue electrolytic medium as shown here. Now, there is no macroscopic galvanic cell here we have an electrolyte medium and there are two phases. For now I will assume that the phase A behaves is more an anode as compared to phase B. And therefore, in the presence of an electrolyte it will actually give rise to an electron this electron of course, will combine with the metal in the electrolyte and deposit at the cathode. So, this will be the cathodic reaction while the anodic reaction is that of the solution. So, where is this electro chemical or galvanic cell forming this galvanic cell forming at the microstructural level it is forming between two phases in the microstructure the phase A and the phase B. And in the of course, in the presence of an electrolyte. So, it is very clear that I am talking about corrosion this kind of a corrosion could actually be much more detrimental. Because, now this is occurring at a microscopic scale because phase A for instance could be of the order of say 10 microns it is not a something or 5 micron which is not visible to the naked eye. And since the corrosion is extremely localized it could actually lead to a stress concentrator or could actually lead to a local damage which is not detected and can lead to the failure of the component. Therefore, in many circumstances such kind of very localized corrosion is more detrimental than the uniform corrosion which can actually be planned for. And for which you can actually even plan a replacement or what is called a preventive maintenance replacement. But, can we have electrochemical corrosion occurring because of residual stresses the answer is yes. And in this case what happens is that regions which are tensile in the material and the diagram you can see below here there is a region for instance we considered the precipitate here. And we saw that there are regions which are tensile as compared to the remaining region which could be compressive. So, for instance this is now my compressive region and this is my tensile region. So, suppose I just take as a schematic example a region which is tensile here and a region which is compressive here. Now, you can actually form an electrochemical or galvanic cell between the tensile and compressive regions the tensile region may play the role of an anode here. Where in the metal gives rise to M plus and an electron that means it dissolves into the electrolyte. And at the cathode the positive ion combines with the electron to deposit there. In other words now I can have a selective dissolution of the anodic tensile region. This is occurring of course at the scale of length scale of the microstructure I am showing here. But, it is occurring because not because of course it could be because of the as we saw the precipitate in the previous case. But, it could also occur purely because of the residual stresses present in the material. Of course, we are just taken for say certain distribution of residual stresses here, but they could be associated with for instance a thermal mismatch or it could be across an interface or it could be even be dislocations etcetera. But, the important point is that now selective dissolution of the anode or the anodic region is the region which is tensile in nature and not a different phase or not a different electrode as one normally concedes in a macroscopic galvanic corrosion scenario. Therefore, clearly from these two examples which we have considered now one being the case of heteropetaxial growth of islands. And more importantly heteropetaxial growth of buried islands of the second layer. And the case of galvanic corrosion that residual stresses can play a very important role and they need to be actually elevated to the level of phases in the distribution or of the level of defects in the distribution. And therefore, a comprehensive definition of microstructure should involve phases defects and residual stresses. And of course, the distributions and these two examples are very very important in exemplifying this important aspect. So, Anil has a question. Good question here. So, in the case we are just talked about we talked about epitaxial islands. But, it is not necessary that islands be epitaxial for instance I could take a glass substrate and I could grow a crystalline layer on it which could be for instance a gold layer on top of it. And this gold layer of course, may form some kind of an what you may call as spherical body or a body which is in a certain way you know deposited on this. There is clearly an absence of any epitaxial stresses. Because there is the whole origin of stresses in this case is the need for the system to match atoms or match atomic planes from the epitaxial over layer with the substrate. Why does the system want to do this? The system wants to do this because if there is an epitaxial interface or a coherent interface the interfacial energy cost for the system is small. And at the nucleation stage we know from the nucleation theory that interfacial energy is playing a very important role in nucleation state. And therefore, when the substrate is small it is nucleating on the surface it wants to choose that configuration which can minimize the interfacial energy. But, what is the cost it pays the system pays the cost it pays in terms of the coherency stresses. So, there is a volumetric term which is the coherency stresses which is the additive term in other words it is opposing my nucleation here. But, there is an interfacial energy term which is low and therefore, it prefers to have a coherent interface. And later on when the system evolves with time you will notice that what might happen is that there may be interfacial misfit dislocations which form which actually partially relief my coherency stresses or the epitaxial stresses. And at even larger growth you may notice that this interface may actually become completely incoherent because of presence of many dislocations. So, in a case there is epitaxial system that means for the glass substrate then this kind of provincial alignment is not expected unless of course, again if there is some epitaxy between the over layer and the fill or there is some other origin of residual stresses which can give rise to this kind of an alignment. Of course, this kind of an alignment can be preferred larger the anisotropy of the system in other words the over layers which are grown on. Now, let us talk about another important point with respect to residual stresses in a material. And this comes from often from the engineering application of materials that we think we see that whenever there are residual stresses there is a warping of the component. And often there is a feeling that this is this implies that residual stresses are harmful for the material. But, we also know from a process like short peening that actually that there can be beneficial effect for residual stresses. So, we should understand that residual stresses can be beneficial or detrimental depending on the situation. So, we will take an example of both. So, let us talk about you know when you talking about stress corrosion cracking which is an accelerated form of corrosion in the presence of internal stresses in the material or the component in this case clearly residual stresses are bad. So, because that is what is giving rise to this what you might call stress corrosion cracking. And in the absence of stresses this will be normal corrosion and that will be much which will have a much lower impact or the negative impact on the material. So, clearly there are reasons to believe that residual stresses can be bad like in the case of stress corrosion capping or warpage of a component etcetera. But, there are good number of examples which we can talk about in what you call to illustrate the beneficial effect of residual stresses and we can consider a couple of them in the current slide. When we are talking about transformation toughened zirconium. So, what happens in this case that the crack tip stresses and we saw that crack tip stresses themselves of course, crack tips have no stresses, but they tend to amplify far field stresses. So, we have some sort of way co-opted crack tips into the definition of residual stresses. So, the crack tip stresses which now amplify the mean field far stresses and this high stress value leads to the transformation of cubic zirconium to tetragonal zirconium. Because, in transformation toughened zirconium the zirconium is present in the metastable form and because of the presence of these stresses the zirconium transforms from the cubic form to the tetragonal form and often also to the monoclinic form. And this leads to an increase in volume associated with the phase transformation this increased volume imposes a compressive stress on the crack tip. So, cracks can only open in the presence of tensile stresses. So, I am applying a mean field far field stress and ahead of the crack tip in a material like this for instance my material I see that the crack tip can actually amplify my stresses I am now assuming a sharp crack. Now, if the region ahead of the crack tip is actually transforming say suppose this region because of this high presence of high stress actually transforms. So, this is my region of transformation. So, this volume is larger because of the presence of this tetragonal zirconium and monoclinic zirconium and this imposes a kind of compressive or closing stress on the crack tip. Thus what I observe is that this material actually gets toughened. Because, if there were no transformation toughening zirconium being a brittle material this or ceramic the crack would propagate and lead to failure of the material. But, because of this phase transformation you can see that this is actually giving rise to toughening of the material. So, this is some kind of a secondary effect coming from crack tip stresses, but a direct effect is the next one which are going to consider which is the example of toughened glass. So, we know that glass is an amorphous material and here we are talking about silicate glass and normal glasses which we have for in window panes etcetera. And now because being a very brittle material in spite of its beautiful transparency etcetera it has certain limited applications. Now, we also know that there are glasses like many in many cases which are actually impact resistance we can talk about bullet proof glasses etcetera. So, what is the origin of the toughness of these forms of glasses and we will consider one way of actually toughening gases which is by putting residual stresses into this glass. So, what happens is that let me consider a molten pool of glass here. So, I can allow this molten pool of glass to solidify uniformly, but instead of doing that what I do is that I blow cold air from the outside. And because of a thin crust of glass first solidifies and after this of course, I stop blowing and allow my entire glass volume to form a solid state. Of course, we should note that glass being amorphous it does not display a strong melting point, but nevertheless its viscosity increases to a certain level above may be about 10 power 12 points where it can be considered as a solid. Now, what happens when the remaining of the pool of the glass tries to solidify it will contract, because the liquid has the solid has a lower volume as compared to the liquid. And therefore, when it is trying to contract this glass as its solidifying the outer regions or the upper outer crust is actually already solidified. Therefore, what will happen it will try to pull in this region and thus impose compressive residual stresses on the surface. So, if I look at the residual stress plot I would notice that the close to the surface you will actually have a large compressive stresses and say and this is a negative side compressive side and of course, they will be a small. So, if I plot myself residual stresses we have a large compressive residual stress on the surface which is which will stop actually crack propagation from the surfaces. And we are talking about glasses we should note that cracks on the surface are twice as deleterious as cracks on the interior. And whenever of course, your body is try to be deformed in bending the maximum stresses occur on the surface. Therefore, from two points of view from the fact that in certain kinds of loading the maximum stresses occur on the surface number two from the point of view that actually cracks on the surfaces twice as deleterious as cracks on the interior. So, suppose I had a crack of length a on the surface this would be as bad as a crack of length 2 a on the interior. Therefore, I need to keep my surface as compressively loaded as possible. And that is what is being done by this process of selectively cooling this and solidifying the surface first then cooling the remaining of the interior to give rise to a glass. So, the composition has not been change in this case the phase distribution has not been change in this case what has barely be we have played around with is the presence of residual stresses. And as we know that residual stresses always 0 overall in the body given by the formula integral sigma d v equal to 0. Therefore, the outer is in compressive stresses there is a small residual internal tensile stresses as well in the middle of the body. But, this large compressive residual stresses actually gives rise to a large strengthening or toughening of the glass which can be three to four times or even higher in many cases. Therefore, now playing or engineering my residual stresses I am able to obtain large toughness in the glass. Therefore, it is very clear that residual stresses are very very important and they have to be considered as a part of a functional definition of microstructure. And as we will see when we go into the study of nano materials we will note that each one of these components of the definition of a microstructure. For instance the phases or the defects defect distribution or defect structures and the residual stress all could get into the nano scale. And therefore, in nano materials all of them would have a equally important or if not a more important role to play in determining the properties of the of course, the nano structure or the nano material and also the entire component as a whole which is made from such a structure or material. Now, we could of course, talk about a certain length scale we could talk about properties at that length scale. But, more important is that how we go from one length scale to another length scale to even larger length scale. And finally, how each length scale talks to each other giving rise to what we may call what we may usually observe as a macroscopic property. So, in other words it is not f enough that I reside in a certain length scale and worry about properties at that length scale. I need to see how this length scale talks to next higher length scale, that length scale talks to next higher length scale and how our properties varying across this length scale finally, giving rise to what we may think as a property or the performance of an entire component. So, just to take up a couple of examples that how we can talk about what you may do, you may do a traversing of what you may across length scales. And we may take up two simple commons examples and we will talk about all the differenting interesting aspects which come about when you do a traversing of length scale. So, we talk about polycrystalline copper as a first example and as a second example, we will take up the example of what you call an ion sample which has been magnetized. So, we will take up two examples to understand how we are going to traverse across length scale. So, what is the first length scale I need to talk about in the case of an copper polycrystalline. The first length scale is of course, the atomic length scale and if you note that the atomic length scale the important thing is that at any finite temperatures atoms are actually vibrate, they are not stationary. That means there is lot of disorder at the fundamental atomic length scale. Therefore, at the atomic level there is order only in the average sense and any temperature and typically we are always talking about temperature which is greater than 0 Kelvin. Atoms are constantly vibrating about the mean lattice position and of course, we not yet come to the lattice scale, but they are vibrating about some kind of a mean position. Hence, in a stricter sense perfect order is missing. At the next length scale which is the length scale of the what you may call the unit cell length scale level, the atomic arrangement becomes evident that is the crystalline crystal structure develops. So, here I have divided the problem into four length scales. The atomic level which is at the angstrom length scale, the unit cell level and if I am talking about simple crystal structures like copper I have to talk only about a few angstroms, but there could be other crystal structures which have a much larger lattice parameter and they could be in the nanometer length scale. But typically it is between about a few angstroms to a few nanometers. Typically though we could always talk about crystals which have much larger lattice parameter. The next level which I am talking about which I will talk about is the level of the grain or the level of the single crystalline part of a polycrystalline specimen and typically this is of the order of microns. Of course, special processing we can see that these grain size can actually be reduced to nanometers which is one of the things we will be considering during this course. How can I produce nano crystalline copper and what are the importance or advantages of producing nano crystalline materials or what you may call nano structured materials. When you jump up from the micron scale to the scale of the material or the scale of the entire component, now we are talking about centimeters and even larger. Then we shall see how the properties have come a few orders of magnitude in terms of dimensions from angstrom to a centimeter which is about 10 power 7 orders of magnitude and how I can actually see various kind of phenomena flipping from like I am talking about order, how order flips from a highly disordered state or average order sense to certain other kind of configurations. So, at the unit cell level where the atomic arrangement becomes evident to us that means we are actually seeing a crystal structure develop concepts like burgers vector emerge. So, when I am talking about a burgers vector I am actually using my unit cell as the fundamental definition and it is a burgers vector is nothing, but a fundamental lattice translation vector or the shortest lattice translation vector for a perfect dislocation. It is at this level I actually do an averaging with respect to say probabilistic occupation. For instance suppose I am talking about a disordered alloy which is say for instance a nickel 50 aluminum 50 alloy and that means that there are 50 percent of the atoms in the system which are of the type of nickel and 50 percent of the type of aluminum. And of course, for the simplicity let me talk about a B 2's kind of a structure where in the body centers there is a body center and there is an phase center. And we are not limiting ourselves to the example of polycrystalline copper we are actually digressing a little bit to an alternate example just to tell us that where does the averaging with respect to composition take place it takes place at this level of the unit cell. So, in an NAAL system with 50 50 possibility I if I take a single unit cell I would may notice that it may not contain 50 50 percent atoms of nickel or aluminum, but when I take a few unit cells or larger cluster of unit cells and average out there then I would notice that 50 percent of the atoms are nickel and 50 percent of the atoms are of aluminum. And if it is a disordered alloy then I would notice that that there is no preferential occupation of either aluminum or nickel in any of the sites. So, it could be randomly pre-position anywhere. So, first I have the level of the atomic scale where in there is no perfect order there is actually atoms are vibrating about their mean position when I go to the next length scale with the length scale of the unit cell. Again if of course, I am talking about a pure material I would notice that there is some sort of a crystalline order that means, atoms are position close to the lattice position. And in other words I could not worry about these small oscillations which are present at the atomic length scale. And if I am talking about a few unit cells then I may see that these oscillations are much smaller than the what you may call a few lattice parameters length scale. But also I notice that at this length scale there is an there could be disorder of a different type the disorder with respect to for instance a probabilistic occupation of atoms or the compositional averaging which I need to do. The next level which is not just the length scale of a few unit cells, but the length scale of a single grain. And you can see in this microstructure here which is shown for a pure copper that there are many grains within these grains you may observe interesting features like these are twin boundaries here. And I could restrict myself to this bright region which is a single grain of copper. Now the single grain of copper of course, is randomly oriented with respect to its neighboring grains. And there is no preferential alignment I hope with respect to the neighboring grains. At this grain level which is a single crystalline part of this polycrystalline material there is nearly perfect order. Because the scale of atomic vibrations is too small compared to the scale of the entire grain. The scale of the entire grain could be in the order of for instance it is about 100 microns. While the atomic vibration is less than as lattice parameter. Therefore, at this length scale I can think of the system as being nearly perfect with respect to my thermal vibration or the disordering which is coming from thermal vibration. However, at this length scale I may observe that there are crystallographic defects like there could be dislocations, there could be vacancies, there could be micro cracks etcetera. And there could be even stacking faults or as in the case we are seen here a twin boundary. Therefore, even though with respect to the thermal vibration there is some sort of an order at this length scale. But there is no perfect order even at this length scale from the perspective of defects like crystallographic defects in the material. At this scale the material is also anisotropic for instance it could be anisotropic with respect to elastic stiffness. And if I were to talk about the elastic properties of this body actually I will have to feed in three independent elastic moduli which are typically called E 1 1, E 1 2 and E 4 4 if I want to capture the elastic behavior of the material at this length scale. Therefore, the material is anisotropic that means there is a direction dependent to its properties at the length scale of a grain. However, the same single crystal could also be isotropic with respect to certain other properties for instance with respect to its optical properties. But definitely it is anisotropic with respect to its elastic properties. Next length scale which we need to talk about is for instance a length scale of the entire sample. For instance here we have seen a photo of a copper sample. At this length scale you will notice that the material is actually isotropic. And the reason it is isotropic is that because the grains which are anisotropic are now randomly oriented in space. And therefore, when I am averaging over all these random orientations and for we see that there is no preferred orientation to this elastic moduli and I can describe such a material in terms of just two moduli that is E and nu or of course, I could use G and nu or one of the other two moduli. Therefore, I just need two elastic moduli to describe entire polycrystalline copper specimen. So, there is no preferred orientation at this length scale. However, if I roll or extrude this material I would notice that I can develop a texture in this material. In other words there could it could actually develop a preferred orientation now and which occurs due to partial reorientation of the grains. And of course, the kind of texture which develops depends on the deformation process which has been employed. But now in the presence of texture the material again develops some of the anisotropy. In other words we have retrieved some of the anisotropy which was present at the grain level. So, it is one of the crystallographic anisotropies which was present which you actually we have retrieved. Therefore, we can see that concepts often get inverted as we go from one length scale to another and how properties change actually go from one length scale to the other and therefore, it is very important that I understand my material behavior or material properties material structure by traversing across entire length scales. To summarize this length this slide we have just now seen we will notice that when we are at the length scale of the individual atom there is lot of disorder there is thermal vibration. At the length scale of a few unit cells we see that crystalline order has developed. And when I am talking about a say few tens of unit cells the atomic vibration is much smaller compared to that length scale. And at the definitely at the grain length scale I see that things are completely ordered with respect to this thermal vibration. However, at the grain length scale we would notice there are other defects which come into play which are crystallographic defects like dislocations voids etcetera or even stacking faults or twins as the photo micrograph shows here. And therefore, again I am dealing with certain other kind of disorder at this length scale. Now, if I am talking about the single unit cell level I would notice that my system has to be described by anisotropic properties which is what is seen at the single grain level. But, when I go to an entire component level I see that now my material behaves isotropically unless of course, it has some kind of a texture. So, a material typically with randomly oriented grains is completely isotropic. And at this length scale I do have to worry about some amount about the crystallographic defects in the present in the material. But, more importantly also I have to worry about the defects which arise between the grains for instance now I need to worry about the grain boundary in the material. So, as I am going to different different length scales I worry about different issues which finally affect my properties. Now, for instance the presence of grain boundaries would imply that if I am talking about a single crystals and a presence of dislocations the material will be weak to a certain extent. But, suppose these dislocations want to move grain boundaries would typically provide an impediment to the motion of dislocations. And therefore, we have hardening which we often refer to as the hall patch kind of a hardening. At the entire material level now I have only to describe the whole properties for instance elastic property which is 2 independent moduli unlike the case of the grain wherein I have to use or a single crystal where I have to use 3 independent moduli. So, it is clear that when I want to understand any property I want to understand structure I want to therefore, deduce performance of a component which is made from a material like shown here I have to traverse across length scales. It is not a essential that I it is not enough that I worry about only one length scale. And of course, I have to talk about various processing which gives me as to a component the effect of those processing and the processing parameters on the evolution of the material at these various length scales. For instance suppose I am doing a heat treatment that not certain temperatures for instance the material if I am talking about for instance a solid solution of nickel and aluminium it could be disordered, but certain other temperature it could actually get ordered typically it would get ordered lower temperatures. And this ordering can actually alter my crystal structure and if it orders my crystal structure it is going to alter my properties my concepts like burgers vector and it is also going to alter my entire set of properties which arise in length scales above the crystalline level. Now we are in a position to actually proceed to the second example which is the example of an ion sample which has been magnetized.