 In this lecture we will learn about the deformation behavior of nanomaterials. In this world we always worry about nanotechnology, it means we need to synthesize nanomaterials into some useful shapes. So, that is the reason we need to know exactly what will happen once we start deforming a material and make it into some useful shapes. So, that is why the deformation behavior of nanomaterials is a very key component in devising nanotechnology. So, we need to learn what is the science, what is the mechanism through which nanomaterials can respond to certain mechanical stress. Also, there had been very contrasting theories related to nanomaterials because it is very difficult to exactly pinpoint what is happening at that scale. Again, depending from material to material say from a very ductile copper to say ceramic materials, the method of deformation can be very different and again with varying grain sizes. Generally, a material will not have a uniform grain size, it might have a variation of grain sizes starting from nanometers to say micrometers. So, the overall deformation mechanism in nano grains will be very different than that of micro grains and also the interplay between nano and micro can also lead to very much difference in the mechanical property response. So, that is why people generally research is defined very contrasting properties in terms of the way nanomaterials behave. So, there is a consistency in terms of developing a material with a consistent grain size and then advancing what is exactly happening at that particular scale is very very important. So, there are couple of theories which have basically come across. So, starting from the very well known hall pitch relationship and the hall pitch relationship is given by a sigma equal to sigma naught k d power minus 1 by 2 or k under root d, where sigma is the yield stress, sigma naught is the friction stress needed to move the individual dislocations, k is the material constant and d is the grain size. So, we can see the yield stress will start increasing as we have lower and lower grain size. So, that is the basic method or basic relationship between the yield stress and the grain size of the material. So, we can definitely find that when we have micro grains, we have d which is pretty high and then when we have nano grains, when d is pretty low then we can see when we have nano grains, our ultimate yield stress will be very very high. So, that is why people now start towards manufacturing bulk nanostructured materials. Again in this relationship though it is a continuous relationship, we can see that that once we have sigma and we have grain size. So, we can see the grains as we have, this side is nothing but higher grain size. So, we can see as we go for a lower grain size, our yield stress starts increasing, but after certain point what happens is that it starts decreasing at certain point as well. So, according to this relationship, haulage relationship, the stress should keep increasing as we go along. So, this should keep increasing with respect to decreasing grain size, this is the grain size along this side along x axis, along y axis we have the yield stress. So, this should keep increasing as we lower the grain size, but after certain point we start decreasing the yield stress and this region is called inverse haulage and this is the haulage relationship. This also is a region of concern as we go along I will talk about it. So, this haulage relationship talks about that sigma equal to sigma naught k d power minus 1 by 2. So, as we start decreasing grain size, we should see a increase in the yield stress. There are three regions which are of concern to us right now. We can see that region 1, region 1 is the, it is a region where we have well behavior of haulage description. So, the haulage relation is maintained in region 1 that is for single crystals to a grain size of about a micrometer. So, when we have grain size pretty high from a single crystal to approximately say 1 micrometer, the haulage relationship is very much valid that it means that the sigma is really dependent on the d power minus 1 by 2. So, as we start decreasing the grain size, we will see an increase in the yield stress. In region 2, what is happening is starting from 1 micrometer to say approximately around 30 nanometers. So, for the grain sizes which are ranging from around 1 micrometer to about 30 nanometer, haulage relationship is approximately or roughly holding it, but again it deviates from the classical exponent. So, in this case our exponent of sigma equal to sigma naught plus k d power minus 1 by 2. So, this exponent instead of minus 0.5, it comes out to be approximately near 0. So, that is one more plausible explanation that because we will come to that later, but what we can see here is that as we are reducing the grain size, we are seeing that exponent is now coming to approximately 0. So, that is what we can see that slope is approximately along this side. Again if we go much below than that, so below a small critical grain size. So, we can again see that our k itself the constant itself can be 0 or where the strength actually decreases. So, what is happening is our k is also going to 0 and also the strength is actually decreasing as we decrease the grain size. So, we are talking about this region. So, for this region in this case we can see that there is a decrease in the yield stress as we are lowering the grain size. In this region we have very well maintenance of the haulage relationship. So, in this case we do not worry much about it because it is well following the haulage relationship and then this and the second region where we have grain size between 1 micro meter to 30 nanometer. The overall exponent of minus 0.5 is basically coming going to approximately 0 and third case also we can see the slope itself also is going to 0 it is not k I guess it is slope. The slope is from minus 1.5 it is not going to same, but it is going to 0 itself in the third case as well. So, we can see the haulage relationship can be described into three regions. One region where we have very well maintenance of validation of haulage relation. The second case the k the exponent is approximately 0 and the third case we have again decreasing value of yield stress as we further lower the grain size and that limit is approximately around 15 to 30 nanometer and it can depend from material to material. So, what is the role of grain size? So, there are basically two models which describe the role of grain size. So, we can see once we have a large grain size it incorporates the dislocation pile up model it means it is associates dislocation. It means pile up occurs due to presence of grain boundary and grain boundaries act as a barrier from where dislocation pile up is accommodated and that in turn will activate the dislocation sources in the nearby grains. So, once we have a grain boundary it is acting as barrier to dislocation pile up. So, therefore, what will happen it will start inducing some stress concentrations and that will activate the dislocations or sources in the nearby grains. So, in that in that case it will start initiating slip from grain to grain. So, once grain will start slipping it will again lead to the deformation of the grains, but at much higher stress level. Dislocation pile up theory is applicable to nanomaterials provided that there cannot be high number of dislocations within a grain. And what they will do is they will limit the mean free path for the dislocation. So, as soon as we have more and more grain boundaries it will start decreasing the distance between the dislocation and that basically will induce strain hardening because now we have more number of dislocations per unit area. So, we can see there are two different approaches for large grain size it will act as a barrier to dislocation pile up and that will induce some dislocation sources in the nearby grains and that will initiate the slip from grain to grain, but at much higher stress level. And secondly it is the from the grain boundaries which act also act as a dislocation barrier and they will limit the mean free path of dislocation and in turn it will also induce strain hardening. So, that is the way the role of grain size has been defined by two contrasting cases. So, that is again leading to the some additional stress that is being generated because of the presence of grain boundaries. So, we can see in a nano crystalline material we have more number of grain boundaries or a very high grain boundary area. So, that in turn will generate that will initiate much more of dislocation sources in the nearby grains as well even if it is acting as a dislocation pile up. So, it will also induce some stresses it will limit the mean free path of the dislocation and it will again require much more higher stress to pass along the to cover the obstacle of grain. So, in turn will have will observe much strain hardening in this case, but again there is some problems with the pile up model. So, dislocation pile up though it is been said that dislocation start to pile up near the grain boundary to initiate the stress or the enhance the strain hardening, but again the pile up model we cannot use as it is for both BCC as well as FCC because one more thing is that it is it has been told that there is a linear dependence of stress on d power minus 1 by 2, but it is only for large number of dislocation pile ups. So, what happens is that this model is valid only for grain size approximately till 1 micrometer. So, what will happen once the grain grain sizes much below 1 micrometer, we do not have enough number of dislocations pile up which can occur in a material. So, this pile up model does not become applicable to grains which are of smaller sizes of much lesser than 1 micrometer, but still we see increase in the yield stress which is which is responsible responsible for enhanced strength of nano crystalline material. So, this pile up model though we keep saying the grain boundaries are acting as a resistors or as a pinning agents for dislocation pile up, but that does not hold true for nano crystalline material because first of all this pile up model though it is much well accepted it cannot be utilized for both FCC or BCC type of pile ups as it is. And secondly there is a linear dependence of stress on d power minus 1 by 2, but only for large number of dislocation pile ups. So, as we go about reducing the grain size what will happen we will have only very few dislocations that can occur in a nano crystalline material. So, they do not become applicable to grains of much smaller size because what is happening here is number of dislocations grain size cannot be very large and the overall assumption here is that we have a large number of dislocations pile ups. So, these two are very contrary in nature for nano crystalline as well as micro crystalline. So, this pile up model basically cannot really describe what is happening at the nano crystalline regime. So, we always require some additional terms for the fine grain material. So, that is why we can see that what is applicable for a coarse grain material does not hold true for a fine grain material as well. So, we need to incorporate some additional terms which can take care of this strengthening in the nano crystalline regime. So, these are again the limitations of this pile up model. So, though we are saying that the grains are being responsible for inducing the strain hardening by either inducing the dislocation nucleation sites in the other grains or initiating nucleation of dislocation sources at other grains or by inducing the pile up by occurring as a barrier to the dislocation. So, the overall mean free path of dislocations will reduce and that will induce strain hardening, but that requires that the number of dislocations dislocations are pretty high and though that can occur in micro crystalline material that may not hold true for nano crystalline material because in a nano grain we might have very limited region that may allows a 1 or 2 or 3 or tens of dislocations and not more. So, in that term we require some additional terms to describe a very fine grain material and also what is what is what is observing what is being observed in a lieu of that is the dislocation pile up mechanism is not so strong for a very fine grain size or a nanometer size grains. So, in term we will see only 1, 2 or may be 5 or dislocation in the pile up. So, ideally we should not see any further increase in the yield stress and also for grain sizes which are very less than say 10, 15 to 30 nanometer we will see a inverse hall patch relationship it means the overall strength reduces as we increase reduce the grain size. So, that is again one more finding of this inverse hall patch relationship that there are not there the everything is kind of scattered. So, there are no as no grains available for inducing that dislocation and so there thereby we can see an inverse hall patch relationship with grain size less than 15 to 30 nanometer or so because now there are only 1 or 2 dislocation which can basically pile up and much below that even one dislocation may find very hard to go into there and there is no further increase in the yield stress or eventually your yield stress basically decreases as we reduce the grain size that thing is being given as a inverse hall patch relationship. So, essentially new descriptions are required to basically model what is happening at the because of grain boundary. So, hall patch model is now being said that it is acting more on the grain boundaries rather than on dislocation. So, once we talk about dislocations we say about dislocation pile up or generating new sources of dislocation, but now right now we can say that hall patch model is now more on grain boundaries rather than the dislocations and it is found that the overall operation of this one is very similar to that of cobalt creep, but in this case what is what is the case is the overall stress is proportional to the d cube. So, what is happening here is so we also say that there is some stress which is which is responsible for the which is responsible for the deformation of this material with respect to d cube. So, what we can see here is that our overall constant that b is not dependent on strain rate, Boltzmann constant absolute temperature, c is the proportionality constant, omega is the activation volume, delta d is the width of the diffusing channel or the grain boundary and d g b is the diffusion constant for grain boundary. So, we can see the slip can also occur along the grain boundaries and that is responsible for the overall stress that is being induced by the material for its further deformation. So, but in this case what we are seeing here is that sigma has a dependence of d cube. So, the steep fall of sigma versus d minus 1 by 2 is not observed experimentally that is what so that is what we can say here is. So, for very fine grain sizes the overall sigma versus d power minus 1 by 2 curve it is not exactly been observed experimentally. So, again it is further needs that though we can incorporate this cobalt creep into the model, it requires some more modification for encompassing the decrease that occurs below certain grain size and also the relationship of d power minus 1 by 2 only in the micro crystalline regime and then approximately a constant value of this exponent along certain grain size limits. So, that is to incorporate that this was the earlier model which basically came into existence by comparing the cobalt creep or the grain boundary diffusion along for encompassing the for encompassing this deformation. So, again it again we can see that the polycrystals with very large grain size they will obey hall pitch relationship when the grain size is approximately till 1 micrometer, but what is happening is for very smaller grain size much smaller than 1 micrometer this cobalt creep is active. That is why we had this earlier term we had this earlier dependence of this stress on the cobalt creep. So, we can define the stress required for cobalt creep and it can basically occur at very low stresses or very low strain rates or even at little higher temperature. So, these are the certain limitations for this one. So, we can also see that the deformation can occur easily for nano crystalline materials for even the temperatures can be much lower as compared to that of a conventional material. So, again this can be encompassed. So, for smaller grain size we can see cobalt creep is pretty much active that is responsible for the extensive plasticity of this nano crystalline materials. So, that also need to be considered, but what is the relationship between those two will come to that like what is essentially leading to the enhanced stress yield stress required for the deformation. So, we will see that part. So, for poly crystals we can see that they are obeying hall pitch relationship approximately till 1 micrometer, but for smaller grain size cobalt creep also becomes very highly active. It means the overall dependence of stress on d cube is also very much it also needs to be considered. Apart from that what can also happen the bulk diffusion can also occur via neighbor hiring creep, but this neighbor hiring creep is basically very insignificant because now our constant for diffusion constant for grain boundaries very very higher in comparison to that of the diffusion constant for bulk diffusion. So, we can see our d g b is very very higher than the d that of bulk. So, we can always ignore this neighbor hiring creep that is for the bulk we can always ignore it and we can always consider the cobalt creep or grain boundary diffusion. So, because in nano crystalline materials we can see that the grain boundary area is also very very high it is it has diffusion constant also is very very high in comparison to that of d bulk. So, we can always ignore this neighbor hiring creep. So, overall the cobalt creep contribution can be modified as sigma is equal to b naught which is a different constant divided by d plus b d cube. Now, this is much more seen just much more similar to that of what is seen experimentally. So, we can see for very fine grains we can consider cobalt creep we can utilize a modified form of the cobalt creep as the stress required for cobalt creep is equal to b naught by d plus b d cube. So, the net yield stress for deformation can come from the contribution both of dislocations the grain boundary dislocations are also for the cobalt creep. So, what we can see what we can see is the overall stress which is required is not dependent on sigma naught plus k d power minus 1 by 2 that is for what is required for the there is a nothing but the hall patch relationship. And now there are some additional terms which come into picture because of the cobalt creep and this b naught d everything are nothing but material constant and we can define certain ways of estimating them as well. So, we can see that this second term becomes very very much significant we need to incorporate term. So, we can see for nano materials last three terms are significant for nano material because now the cobalt creep also is very very much predominant for cobalt creep last two terms are predominant. So, we can see this k 1 is the constant for a particular material for nano material for cobalt creep last two terms are very much important for grain boundary dislocations first two terms are very much important. So, what we can see because of dislocation pile up or grain boundary dislocation we can see that the resistance is coming from the grain boundaries. So, these become very very much significant last two terms are from the cobalt creep deformation. So, again we can compare these two terms and then we can say approximately we can define a region where we can see the relative contribution from both of them. So, we can approximately estimate when the decrement of this stress can occur because in this case we can see that stress is directly proportional to the d cube. So, in when we have nano crystalline material the overall dependence of this strain basically now starts reducing as we have finer than finer grain size because stress becomes very much easier when we have certain grain size. So, initially we had a grain dependence of d power minus 1 by 2, but at certain region we can see that the sigma is not directly proportional to d cube. So, at certain grain size. So, it means the stress required for deforming a material which is nano crystalline is very very easy when we have when we have predominance of cobalt creep. So, once we can compare these two terms one term because of the grain boundary dislocation and second thing because of the creep that is occurring the cobalt creep which is predominant because of the grain size again. So, we can see that we can find out certain region at which this term will start decreasing. So, we can see that approximately certain point why we have d star. d star means that when we have grain size higher than that we will see strengthening with the reduction in the grain size that is where we have predominance of grain boundary contribution and below that we have contribution from the cobalt creep. So, that is what we can we can identify that we can have the overall contribution from either from the grain boundary dislocations or it can also come from the cobalt creep. So, that is how we can distinguish between what is we can also incorporate the term of reduction in the yield strength yield stress as we are lowering the grain size. Now, though well said about all these three all these three the main thing is we also need to now control the processing of material because that also will affect the deformation behavior of nano materials. So, once we know what is the overall relationship between those two we now we can somehow control the material processing and achieve a material for particular deformation for making into some useful shapes. So, that is why we need to see what is the overall deformation mechanism of this nano crystalline material because now we know that we have some contribution from the grain boundary because of pile up or generation of new sources of dislocation and the ease of deformation because of cobalt creep which is predominant for the nano crystalline material. So, the creep can occur at even at very low temperatures. So, that is what combining these two terms we can see overall dependence on D or the grain size and that dictates that we have certain critical grain size below which we can start seeing decrease in the yield stress. So, again there had been so many control control theories from researchers that many researchers say that the overall plasticity of the nano crystalline material is very very low whereas, other set is very very high and again depending on the processing it can induce some defects into the structure. So, again that might lead to change in the plasticity of the materials. So, there had been very controversial theories from regarding the nano crystalline material which say either they have limited or they have super plastic plasticity available in nano crystalline materials. So, that basically is very very different in different cases because what is happening is strength and plasticity they depend on the specific deformation mechanism. One needs to know what is happening exactly what sort of mechanism is predominant when a certain processing is being done or when a certain deformation is being done. So, the overall plasticity which will result or the strength which will result depends very much on the deformation mechanism which deformation mechanism is predominant in controlling the deformation and thereby leading to the certain strength and plasticity. So, there are many theories which have come about but which one is very convincing is that there is a grain boundary sliding plus grain boundary diffusion and that leads to the plastic deformation of nano crystalline materials. So, the overall plasticity is attributed to the grain boundary sliding and the grain boundary diffusion that leads to the plastic deformation of nano crystalline materials. So, an ductility can be enhanced the overall as we know that the strength can be controlled by the cobalt and the dislocation part or dislocation coming from dislocation pile up coming because of grain boundaries. But again apart from that we also want to increase the ductility. So, strength can be controlled by a grain size the yield stress but again the overall ductility can be controlled. Once we have a flaw free processing that the processing itself does not induce any cracking or any flaw or any inclusion or any impurity into the system. So, we need to have flaw free processing. So, that will avoid limiting any generation of crack nucleating points and also it will avoid any growth instability. Then third part is that we also need to assist the plastic strain while limiting the shear binding. Once the shear binding occurs it is very catastrophic. So, we need to somehow limit the shear binding as well. So, for free flaw free processing we can achieve very cleaner material. So, once we have a cleaner material the overall points for crack nucleation they are highly limited. So, flaw free processing will lead to a very cleaner material. So, in turn we will have very pure materials overall homogeneity of the material is very high and limiting crack nucleation and growth instability that again depends on the kind of strain rate we are providing or the fabrication process that we have selected. So, in that case we need to limit the crack nucleation sites. So, that we can avoid any catastrophic failure and growth instability. So, we need to avoid any stress concentrators in the material by controlling the processing. And again once we are assisting the plastic strain it means we have to enhance the strain hardening or the strain rate sensitivity. So, we can somehow limit the shear binding. So, we have now three problems through which we can enhance the ductility and three solutions also available through which we can achieve this enhance ductility for nano crystalline material. So, we can see for flaw free processing we will get a cleaner material by limiting crack nucleation and growth instability. We can choose a particular process which can avoid the crack nucleation and growth instability. And then also we can enhance the strain hardening or the strain rate sensitivity to limit shear binding. So, thus we can avoid any catastrophic crack propagation or catastrophic failure of the material once we are processing it or once we are synthesizing it. So, that is how we can initiate these three parameters of cleaner material particular process selection. But mainly we can see we ritualize a limiting crack nucleation. So, we need to limit the crack nucleation and growth and also assist plastic strain or limit the shear binding. So, or may be reduced strain hardening. So, we can see there are two aspects of limiting crack nucleation and inducing strain hardening into the material. So, by selecting certain protocols we can somehow achieve this. So, that we can achieve higher ductility. So, strength can be controlled by a grain size and ductility can be achieved by a controlling the limit by limiting the crack nucleation or inducing strain hardening. So, again to achieve those things we there are two phenomena which we need to control. One thing is the grain boundary sliding. So, we can see that grain boundary sliding what it does is it causes strain hardening. So, once you strain harden. So, we can suppress the plastic strain instability. So, we can avoid the formation of shear binds catastrophic shear binds. So, we can see that this grain boundary sliding can limit the catastrophic shear binding on necking that can occur. So, generally what happens is it can also essentially what it is doing is it is also inducing some defects near the grain boundaries not at the grain boundary triple junctions, but at near the near those sides. So, what we can see here is that once we have a grains generally grains are very much hexagonal nature and then we see there are some predominant in a regular material, but what happens in nano crystal materials is now this region is now further defected. It means once we have this sort of arrangement the third will not generate from this point. Now, third will not generate from this point it will start generating from some other point. So, what is happening is now we are avoiding the generation of triple point those triple points will be there, but now we can minimize this is a triple point, but in this case the triple point is out here that should have been here, but now a defect is generated on a nearby point. So, we can see a nearby point we have generated a defect and that is now reducing the overall stress concentration that is occurring in the material and thereby we can reduce the defects exactly at the triple point. So, what is happening here is that we are limiting the triple junctions and we are producing defect in the nearby region. So, that is what we are able to see here what is what it is doing is it is resulting the grain boundary dislocation. So, now we are resulting a grain boundary dislocation and that is resulting the wedge disclimitation or rotational defects. So, what is happening here is initially we should have seen that we should have a grain boundary like this ideally we should have a grain boundary triple junction like this, but now what do we have now we have some region which is like this. So, now in turn what we have done we have now shifted the grain boundary point to certain other region and what we are seeing we are seeing is a grain boundary dislocation and also what is happening is it is also creating some sort of a dipole because of wedge disclimation. So, we can see that the overall grain boundary dislocation the dislocation that the grain boundary had certain orientation. Now, what is happening here is the overall disclimation is now arrived because now we have one sort of dislocation here second sort of dislocation here. So, we can see there is kind of a polarity that can generate between these two points this is called a wedge disclimation. So, we can see there is a kind of wedge disclimation or rotational defect that can arise because of presence of these two points which are the defects near the grain boundary. So, what is happening is it is causing a dipole or a wedge disclimation which are nothing but rotational defects. So, now once we are rotational defects it creates incompatibility with the lattice orientation. So, lattice will have certain orientation and this wedge will start creating some rotational defect. So, what is happening here is now this will cause some pronounced strain hardening and that is much more stronger than that can arise from the pile up of dislocation or dislocation storage at triple junction. So, once we had this dislocation storage at triple junction this point would have become very much weaker, but now the strains are coming because now we have two different points one here and second one here. So, this is causing some rotational defect and that is causing strain hardening because of defect which are much near to the grain boundary not exactly at the triple point and now it can result the sliding of the grain boundary and thereby it can suppress the plastic strain instability. Otherwise what can happen that this point will become so weaker that it can just lead to catastrophic failure, but in the second case now we have some rotational component into the picture and that will cause some wedge disclimination that will come out by wedge disclimination it will induce some strain hardening and lead to the movement of or slipping of this grain boundary. So, the grain boundary will slide. So, that is what we are seeing that the grain boundary has now slide it to some other location we can see this point the grain boundary is slide it from one region to from this point to nearby region. So, we can see the grain boundary sliding that has occurred into the material and now that this is the one which is which is inducing this plane plastic strain instability which is suppressing the plane strain instability because of this grain boundary sliding. So, we can see that the grain boundary sliding is a very essential component to cause strain hardening in the material and it suppresses the plastic strain instability which can lead to catastrophic failure or by producing shear bands. So, that that is what the necking or the shear binding is now being avoided and that is being incurred by a moving this defect areas away from the triple point. So, triple point instead of instead of triple point we have a defect region near the triple triple junction and that causes kind of a polarity between along those two points by causing a dipole or a wedge disclimination. Now that in turn it is a very strong rotational component and that will lead to a formation of formation of some lattice creating some incompatibility in the lattice orientation and that essentially leads to a strain hardening because of this location at those particular points. So, had we had only triple junction that will be very catastrophic because now we have defect located exactly at one point whereas, in the second case we have a rotational component and that will lead to the grain boundary sliding and that is very much responsible for avoiding the planes the plastic strain necking or the shear binding that could have occurred catastrophically. So, grain boundary sliding will assist the deformation in a very gradual fashion, but also with much more strain hardening and this strain hardening is very much higher than that it can come about from pile up of dislocation at the triple junction. So, in that case we can we can achieve very high plasticity via avoiding this shear binding or necking or avoiding this plastic strain instability in the nano crystalline material. So, that is one more advantage of grain boundary sliding the dominance of grain boundary sliding in the nano crystalline materials. So, once you induce some strain hardening because of the grain boundary sliding now this additional stress also has to be now removed gradually. So, it can accommodate more and more straining. So, that is why we have grain boundary diffusion which is also we require it to capture. So, that we can relieve the stresses that are being generated by the or the strain that are being generated by the strain hardening. So, we can see the strain hardening which is been caused by the grain boundary sliding that is to be removed or released by grain boundary diffusion. So, what is happening is now we have generated those wedge disclinations near the triple junctions or the grain boundary near the grain triple triple junction triple junction. So, we need to relieve those dislocations which are nothing but stresses or strain hardening in nano crystalline material. So, what it will do is now once we have those disclinations those are also again very high stress point. So, they can also act as a stress razor and then they can act as a crack nucleation sites. So, we need to suppress those crack nucleation and growth by this grain boundary diffusion. So, once we can allow the grain boundary diffusion to diffuse then it can encompass those regions and in turn it will remove those stress or strain nucleating sites. The strain hardening has been caused by this grain boundary sliding. It also is essential that we release those stresses because otherwise they can turn as a nucleating sites for cracking and its growth. So, we see that the grain boundary diffusion also is very essential component in controlling the deformation that can be incurred by a non nano crystalline materials. So, what we are seeing is the strain hardening which is been caused by grain boundary sliding. Let us to let us to be relieved by the grain boundary diffusion. So, what is happening that what we saw earlier was the wedge disclinations that are arising at near the triple junctions. So, we have region which are also defect which are also which are also defected near the triple triple point. So, we are now releasing the triple points and creating the defect near those regions. So, in turn that can also act as a stress razor or a stress concentrator. So, we need to allow also allow to release those stresses or strains from those regions. So, what we can see is that we can relieve those disclinations stresses or strain hardening in nano crystalline materials via grain boundary diffusion. So, what in turn it will do is it will suppress the crack nucleation and growth because of this grain boundary diffusion. So, it will encompass those regions which are defected and now it can accommodate those it can somehow release those stresses because of grain boundary diffusion and it allow very equilibrium flow of material around those regions. So, the overall grain boundary diffusion is also very much critical in controlling the stress or in terms of removing or releasing those stresses or strain hardening in those materials and that is very significant either at higher temperatures or lower strain. So, once we are deforming a material at either higher temperatures or lower strains the material gets enough time for relax for relaxing itself. So, that is why we can see super plasticity when we have very high temperatures or very low straining rates of deforming the material. So, in this case we can see that the grain boundary sliding has to be compensated via grain boundary diffusion. So, in this case what we are trying we are trying to limit the cracking both initiation and growth and where grain boundary sliding we are limiting the plastic instability or shear binding. So, we can see the two components where grain boundary sliding we are trying to control the plastic instability or shear binding and where grain boundary diffusion we are trying to release the stresses which are being caused by a strain hardening because of grain boundary sliding. And to achieve this grain boundary diffusion we need to work the material at very high temperatures or at lower strain. So, that it becomes easier to release those deceleration stresses or strain hardening in the nano crystalline materials. So, both the components grain boundary sliding and grain boundary diffusion are very important because once grain boundary sliding is inducing certain strain hardening into the material that has to be released. So, the material can accommodate much more deformation or much more working on the same material. But this working has to be done at pretty high temperatures or very low strain very low strains or low strain rates. So, in turn what is happening is it is relieving the disclimination stresses or the wedge disclimination stresses which are inducing a rotational component and leading to this grain boundary sliding. So, at certain level those stresses have to be removed. So, the more working of the material can be done to again without cracking the material or reducing the crack growth. So, that in that fraction grain boundary diffusion also becomes very very important component of that. So, now coming to the final aspect of toughening say in ceramics we can also toughen the material by inducing some nano structured zone. So, we can see that nano structured zone can also act as region where we can arrest the crack. So, once we have a nano crystalline material now the crack when the crack is propagating it will have to encompass very tortuous path. Also the surface area is so high that it can also accommodate some slipping or sliding between the faces or particles. So, thereby the crack can get arrested at the nano structured zone. So, once we have very fine grain structure the crack once it is propagating it can get arrested at those regions. Also it can do is micro crack deflection. So, once we have a crack propagating and once we start deflecting it change the path of the crack itself. Now, the crack is bifurcates and it also requires much higher energy to propagate further because it initially started with a unidirection. So, it wanted to go from left to right, but now once we are deflecting the crack and diverging the crack. Now, crack is propagating in very different directions and internet is losing the energy by creation of new surfaces. Also we can restrain the grain growth. So, as we see that now once we have a very fine grain size we can we can encompass more number of dislocation that may not hold true for ceramics, but by restraining the grain growth. Now, we can allow the grain bounded to act as a barrier for further propagation of crack and in turn it can assist in the toughening of the ceramic. Also we can induce some secondary faces. So, like secondary faces can be much more tougher like in ceramics we can add some nano structured nano structured tubular particles or a non spherical particles and that can act as a regions to absorb the shock. And also we can achieve transformation toughening in some materials like in nano crystalline materials we can see that specifically for zirconia. We can achieve a tetragonal to monoclinic transformation and in nano crystalline material that might be much more easier for the material to undergo this transformation because we are very limited regimes where the material has to transform. So, exactly in front of the crack tip once we are very high stresses the stress transformation can allow tetragonal to monoclinic transformation of zirconia and in turn what is happening the shock is being absorbed because of this particular transformation. And now the crack will experience compressive stresses and then it will lead to the closure of the crack. So, this transformation toughening is occurring occurring via expansion of the volume by say 3 to 4 percent volume expansion. And since crack is now opening up it will start closing down because of the stresses because of the expansion of the material. So, in turn it will lead to transformation toughening. So, we can see again toughening in ceramics can be accommodated by nano structured materials via arresting the crack, via deflection of the crack, by restraining the crack grain growth or by inducing the secondary phase toughening by inducing some nano crystalline materials such as carbon nanotubes or via transformation toughening. And then nano crystalline material once we have a nano particle or nano phase which is present ahead of the crack tap it becomes very easier for the stress to undertake the transformation because these particles nano particles are highly unstable or metastable in nature. So, in that way they can easily transform to the stable phase which is monoclinic and in turn induce induce the volume expansion of approximately 3 to 4 percent and that will eventually lead to the closure of the crack. So, we can see the application of this nano crystalline materials in toughening the ceramics which is a very critical component these days because for structural materials the ceramics are very highly placed. So, in turn we can utilize nano crystalline materials for inducing toughening in the ceramics which is otherwise a highly brittle material. So, this toughening in ceramics can occur via arresting the crack or via deflecting the crack or by restricting the grain growth or by inducing this nano size phases or particles at of the crack tap. So, now because of the instability of this particles they can easily transform to the stable phase via inducing some compressive stresses and thereby closing the crack. So, we can see these particular things are possible because of the nano structured materials and it eases out the toughening which can be achieved in the even in ceramics. So, seeing nano versus micro crystalline grains. So, in what happens in conventional micro crystalline grains is we can get very limited plastic deformation. So, nano crystalline materials first of all we can see once a crack has to propagate it can find easy path of easy path for propagation. There is no presence of any grain boundary sliding and grain boundary diffusion also is very very much limited because the grain boundary area is very much lesser than that of a area of bulk. So, what is happening is in the micro crystalline materials we have very high grain size very large grain size. So, we can find very easy path for crack propagation grain boundary sliding is basically being absent grain boundary diffusion also is very limited because the grain boundary area also is very very lesser than that of a grain. So, in conventional micro crystalline materials what we can see we can see very limited plastic deformation and eventually the material is very very brittle specifically for ceramics otherwise what happens in nano crystalline material is deformation via cracking and shear sliding can also occur along the grain boundary. So, first of all we have nano crystalline material we can see that the for a similar region what we can see the crack path can be very very tortuous the coverage also is very much higher for. So, once we can see when the crack is basically being generated it can it can be very very tortuous by the time it comes out. So, very tortuous crack path crack deflection can also occur very easily because now grain size have very different curvatures. So, deflection of crack path can also occur it can also get restrained or it can also get arrested the crack arrest can also occur because now it is covering much more area it is creating many more new surfaces. So, it can also get absorbed or it can get arrested very easily in a nano crystalline material apart from that it also incurs grain boundary sliding grain boundary diffusion also the grain rotation can also occur. If a particle is not tightly bound to the nearby it can also basically act as a free surface and it can rotate itself and in turn it can accommodate additional energy for its rotation and it can absorb the additional stresses that are been incorporated into the material it can act as a free surface and in turn it can also absorb shock because in this case our grain boundary area is very much higher or comparable to that of grain. So, the phenomenon of grain boundary diffusion also becomes very very much appreciable. So, what we are seeing is the deformation wire cracking and shear sliding along the grain boundaries. So, sliding can also occur between the grain boundaries that makes the material very very ductile because now we have grain boundary sliding we have grain boundary diffusion grain rotation can also occur and also shear sliding along grain boundaries that can also occur. So, in turn what is happening in the nano that we are seeing much more high ductility specifically for ceramics in this case we have very high brittleness for ceramics and the ductility can also enhance for metallic materials as well because of all this phenomena like grain boundary sliding grain boundary diffusion that can also occur very easily in the nano crystalline material. So, this is the this covers the world paradigm of nano versus micro crystalline grain. So, we can see what is the role of nano crystalline material in terms of deformation how it can deform and how it can provide much more toughening to the material both in terms of stresses as well as ductility. So, we can see deformation of nano crystalline material it is being dictated by this hall pressure relationship. So, till 1 micrometer we can see very much validation of hall pressure relationship but starting from 1 to around 30 nanometer we can see that the hall pressure relationship the exponent the d exponent is not exactly minus 0.5 it comes down near to 0 and so we see a some horizontal play to and then apart from that we can also see for grains which are much smaller than 30 nanometer inverse hall pressure relationship can also be very very predominant because in that case. So, apart from hall pressure what we can see that now what is happening is there is something happening at the grain boundary. So, modifications are done to involve some grain boundary phenomena like what is happening because of the cobalt creep. So, the two things one thing is the dislocation pile up can also occur at very lower grain sizes. So, that those are responsible for region for enhancement of the yield stress with reduction in the grain size but at certain critical grain size below that we start seeing decrease in the yield stress with decreasing grain size and that thing is being responsible by the responsible because of that thing is attributed to the cobalt creep. So, now we need to inculcate many different terms. So, we can see yield stress is equal to sigma naught plus k d power minus 1 by 2 which is which is coming out because of the grain boundary or dislocations which are getting piled up plus some constant and then we see components of cobalt creep which now make sure that the smaller the grain size smaller is the yield stress and those things are responsible for reduction in the yield stress as we reduce the grain sizes. So, we can see the dislocation pile up and cobalt creep model can also come along apart from that the toughening or the ductility of nano crystal material is very very high because of grain boundary sliding. Once we have nano crystal material it starts inducing defects near the triple points not exactly the triple points that leads to wedge disclinations and that induces a kind of a dipole or rotational component and in turn it leads to the grain boundary sliding and thereby it the overall deformation can be enhanced because now we do not have stress concentrators at the triple junctions and but those have to be accommodated by grain boundary diffusion to release those strain hardened regions we also require grain boundary diffusion that has to occur for in for encompassing the enhanced plasticity. Apart from that we also see the difference between micro and nano crystalline materials because what is happening in micro crystalline material is we have components of extra energy coming because of the nano phases the ease of transformation and say transformation toughening or also the grain boundary sliding grain boundary diffusion even the grain boundary rotation can also occur and grain boundary slipping or shearing can also occur that induces much more toughening or ductility in the nano crystalline material in compared to that of micro crystalline material and with this I basically with this we complete the overall paradigm of nano difference between nano and the micro components of crystalline materials for for the yield stress or the ductility or the overall deformation of this nano crystalline material. So with this I end my lecture here. Thank you.