 In this lecture, we will learn about nanomechanics. Because of emergence of nanomaterials, it becomes important to learn about what is happening at the nanoscale and to also extract properties, mechanical properties such as hardness, modulus, how does it vary with respect to different nano phases which are present in a bulk nanostructured material. So, in this lecture, we will learn about nanomechanics, specifically about nano indentation in order to extract hardness and Young's modulus out of these materials. Nanomechanics is the field of technology which involves forces to in the order of nanonewtons or displacements which are to the order of nanometers. So, that is why nanomechanics will deal with materials which have a nano size, nanometer size, so that we can specifically design this instrumentation, so that we can precisely measure what are the displacements or the forces which have been incurred by that particular phase which is in the nanometer size range. It also is utilized for fabrication of new materials or scanning a particular property out of it or scanning the morphology or the topography out of it. Also sensing because once we talk about nanotechnology, we also need to sense what sort of forces which are incurring on a particular material. So, in order to comprise the fabrication, scanning, sensing the nanomechanics becomes a very essential tool, learning about nanomechanics. It can again be a non-contact, in that case we can utilize van der Waals forces which is again in the non-contact mode like we can utilize atomic force microscopy and we can utilize the non-contact mode in terms of imaging the topography or it can again be in the contact mode like for nano indentation. So, coming in contact with the material and then extracting the properties or mechanical properties out of it. So, we can see nanomechanics is a very very essential way of learning about what is the response of a nano material. It can be nano phases and how do they incorporate in terms of advancing the bulk properties. So, we can have a nano properties of each and every individual phase plus contribution of some additional forces such as it can be just porosity or say interaction between the two grains which are in nano size. So, that can finally, advance the bulk property. So, once we know what is the response at a nanometer length scale, we can somehow build them up by using a multiline scale hierarchy to achieve a bulk property out of that. So, nanomechanics is a very very essential way means of comprising what is happening at nano scale and predicting what can happen at the bulk scale. And again as we see that it is utilized for fabrication or even scanning or even sensing. Majorly in this lecture we will learn about nano indentation. So, let us see about that. So, nano indentation is utilized for measuring the mechanical properties of thin films, coatings or even smaller volumes. So, see if you have a microstructure and where the phases are in the nano regime and if you want to extract what is happening or what is the change in the mechanical property. Because of those particular phases, we need to see what is the property of individual phases like we have phase A. So, like we have phase A and we have phase B and assuming that the overall length scale is couple of nanometer. So, then once we make a tensile bar specimen, it will give us a property of the bulk. So, we can get a overall property of the bulk using the tensile bar specimen. Which can provide us modulus of the system or stress strain relationship. But if you want to see what is happening, what is the contribution from each and every individual phase like of phase A and phase B. We need to do the nano indentation. So, we can extract the properties out of individual phases. Again if you have a coating, if you have a very thin coating to the order of couple of nanometers of sub micron range, it will say if it is very adhering to the substrate. So, we have a substrate and we have a coating and we want to see what is the response of this coating with respect to the substrate. We might do a tensile bar specimen, tensile bar testing, but we want to see what is the contribution of coating alone. Then it is very hard because even if you do indentation, a micro indentation that might actually have some influence from the substrate also. So, the coating thickness has to be independent because if we are the thickness of this coating has to be independent. It has to be thick enough, so that we do not see any response from the substrate. But that may not be in reality, that may not occur in reality when we have a sub micron size coating on a particular substrate. So, in those cases nano indentation becomes very essential because if you want to extract the properties, the indent or the depth of the indent should be smaller than at least 7 to 10 times of the entire coating thickness. So, that in that way we can avoid the contribution from the substrate and we can get properties only with respect to the coating and not the substrate. So, in order to calculate or measure the mechanical properties of thin films or coatings or even smaller volumes. So, in this case if we have a face A and face B, these are very small volumes because we are talking about grain size which is to the order of couple of nanometers. So, we can individually go to a particular spot and do the indentation at those particular points. So, what we can get is property with the property of that individual grain only otherwise a micro indentation what it will do, its size itself is so big that it will basically go on to. So, if you do a micro indent it might comprise a much bigger regime of indent indent and we get an average property, whereas nano indentation will give us a local property, indent which is micron size will give us a bulk or a average property. So, that is why it becomes very essential to utilize nano indentation and nano indentation through an instrumented indentation it means we are consistently measuring the load which is being incurred on a material as well as measuring the indent depth and that can help us evaluate the hardness as well as the elastic modulus why are consistently measuring the force and displacement. Also we can have a special resolution, so we can also probe the surface using this particular probe tip and we can also achieve a very good special resolution in this case. So, we can measure property which is very very nearby we can measure the properties of the nearby grains that is what is meant by special resolution that is very good special resolution. So, we can measure properties from localized region, so we can have an array of very fine array of points where we can extract the properties and we can get very good properties along the each and every pixel as well. So, that will give us provide as a very good special resolution. So, nano indentation is a very essential tool in terms of extracting the properties mechanical properties of say thin films or coatings or even small volumes using this particular technique via measurement of instrumented via instrumented indentation it means we are consistent continuously measuring the load and the displacement and through this we can also achieve very good special resolution. So, nano indentation is nothing but continuous measurement of force which is to the order of nano Newton to milli Newton, so we can utilize this nano mechanics, so that will have us load range of around nano Newton, but we can go up to micro Newton's or even milli Newton's in case when we require and displacements are to the order of nano meter to it can even go up to micro meter depending on the sensitivity we require from the instrument or kind of the average property which we require from the substrate. So, that is what is basically being attained via nano indentation and it is called nano indentation because the overall depth range are in the order of nano meters couple of hundred it can even go hundreds or thousands of nano meter as well depending on the probe which we utilize or the load which we basically utilize. Again in this case we are measuring load and displacement throughout this process as a load versus displacement curve. So, we have a load just given by p and displacement given by h, so we can directly measure the load as soon as we are loading it the indentation depth keeps increasing and then we can consistently measure the load displacement curve. So, we have load versus displacement in the matter and then while unloading also it can have various responses it can have some elastic and plastic contribution which we will talk about later on through which we can also calculate certain mechanical property such as hardness, Young's modulus we can also do some stress strain studies we can also see what kind of stresses are being incurred while using conosphericality we can also we can also convert the load into stresses and displacement to strain. So, that can provide us what is the stress strain which can being how it can be similar to that what we can achieve from a real engineering test of a bulk sample. We can also utilize time dependent creep measurements either while maintaining the stress constant or seeing the temporal while seeing the temporal response or measuring the strain that can occur or the stress relaxation that can occur at certain load. So, it can also provide a time dependent creep measurements also we can utilize to find fracture toughness it means once we are indenting a ceramic material it will eventually develop some cracking it can be median cracking or it can be radial cracking. So, from that we can always measure what is the toughness of the material. So, that is a semi empirical empirical measure way of measuring the fracture toughness also we can measure the plastic and the elastic energy of sample. So, if we have a load displacement curve we know that this is the elastic response which is being recovered in instantaneously. So, this is nothing but the elastic response and this is the one which is the plastic response as it will recover with time. So, we have loading. So, this part is loading and this part is unloading. So, once we load the sample indentation depth will keep increasing and then as soon as we unload it we can see the unloading part is coming down like this. So, it is loading and then unloading will basically come down not here. So, this much is the strain which has been recovered. So, this much strain is the one which is been recovered. So, it is the elastic part and the one which will recover later or which will remain as it is the plastic part. So, through this we can also calculate the elastic and plastic energies of the sample material. So, we can see nano indentation it utilizes continuous measurement of forces in the range of nano Newton to mill Newton's and the displacements are again in the range of nano meter to the micro meter range, but mainly in the nano meter range that is why we call it nano indentation. And we continuously measure load and displacement to get this particular load in load displacement curve from which we can calculate such as hardness, Young's modulus. We can also convert them into stress strain plot depending on the once we have the area function available or how we can calculate it to the force of the stress and then the strain. We can also utilize to utilize nano indentation to find time displacement creeped measurements by varying temperatures by holding the material at certain stress level and so on or seeing the stress or lessen that can occur. So, we can have many feedback loops associated with this particular testing to calculate these properties. We can also calculate fracture toughness as we said to basically utilize some semi empirical formulas and see what kind of cracking is occurring in the material upon a certain loading is being applied to it and we can also find plastic and elastic energies of the sample. So, this provides the overall potential of this particular testing techniques which is called nano indentation. So, how does the nano indentation will work? So, we have an indentation it can have a Berkovich tip which is a pyramidal tip or wicker step or it can also be conospherical tip which is being impressed on to the sample surface. And what happens is the indentation load and displacement will occur and those are recorded because this is the instrumented indentation and we record the load and the displacement and we see the loading and the unloading cycle that has been recorded in a continuous or even a step wise mode. And the depth starts varying to around several nanometers to it can be it can go even to micrometer. So, as I said earlier that we have a loading and unloading which is occurring and that is being recorded continuously by the instrument and then we can measure the hardness which is given by the load divided by the projected area. So, we can measure the hardness through the measurement of the load and the area. So, through this we can calculate the hardness very easily this is a very normal principle of measuring the hardness. So, the first property is that we utilize a particular indentor it can be Berkovich wicker or it can be conospherical indentor as well that is impressed on a surface. We measure the continuous loading and the unloading out here and right now we do not know what is the area of the indentor which is being impressed. So, it requires certain measurements as we will see later on. So, measurement of area or the contact area is not easy as we can see in a regular bulk testing because the because once we talk about a tip Berkovich or wicker sorry even conospherical tip they may not have a perfect geometry. So, to take care of that we always utilize the standardization or calibration of that particular tip to measure the contact area with respect to the depth of the material. So, that is very much required in any nano indentation instrumentation and the depth is basically varying that is why we need to measure what is the final depth or the depth which is what is the relation of p versus h. So, through this we can calculate the hardness which is obtained by the applied load divided by the projected area. So, we can calculate the hardness using nano indentation. So, a typical pH curve basically will look like that we have loading. So, we have we can see that. So, what do we have? We have an indentor initially. So, we have indentor. So, assuming the geometry is like this and this is a substrate. So, we have this indentor and the substrate. So, initially it will come in contact. So, once we start impressing this indentor on the substrate. So, we have a load we have an indentation depth. So, once we start impressing it, it will start getting into the substrate and it will reach a substrate and this is the indentor. So, we start loading it. So, as the load increases the indentation depth also starts increasing. So, we are we are impressing this indentor on the substrate and it will reach a certain depth. So, that is called h max. This is once it has reached a maximum depth. It is called h max and geometry of this particular the curvature of this one will depend on the material. It can vary from material to material. So, for very brittle materials it might look like this for little softer or some hardening hardened material it might look very different. So, it can have various geometries as it is. So, let us let us take a typical shape something like this. So, we have reached now h max and what will happen and then we start unloading it. So, in the first stage we are loading it and if you want to hold it the load might remain constant. So, in this case we might require to relax the material and certain cases to avoid the viscoelastic effects. So, but right now let us not consider that let us just consider that once we have impress the sample we are unloading it. So, second step that it has it has reached a maximum and then third step we start unloading it. So, in first case we have a impress the sample. So, we have certain maximum depth. So, let us see this is the. So, we have a substrate and this is a indenter and this height is nothing but h max. Now, what happens is we start unloading the sample we start unloading the material. So, what do we have we receive a unloading part like this. So, the slope of this one if we take the slope of this one and while it while the length it has recovered it is called h final this point is called h final. So, what do we have. So, we had h max after unloading we have received which is called h final and then the slope of this curve is given by s that is the compliance part of it. So, we received d p by d h and incorporating this we can we can also see that also the as soon as we the material is recovered instantaneously some to certain h f, but the overall contact depth the overall count because what has happened this depth has been achieved while when the indenter has been recovered at the same time it has also gone under some plastic recovery as well. So, we can see that the elastic just taking the elastic recovery part we can see that ideally it should have gone out here, but so h c is given as h max minus the strain it has incorporated into p max by s. So, actually this is h c and incorporating this particular formula we can see that h c is basically nothing, but it might come up somewhere here. So, what is happening is once we once we indent the material we are seeing the h max or this is the p max and h max is being being achieved here we indent the indent the sample. So, we achieve h max it undergoes the recovery which is which is the final depth which is called h f and elastic part is basically given by h c, but the real h c which is the overall contact that that is being given by h c. So, we have this particular slope which is given by d p by d h elastic part plus the strain which is incurred in the material due to plasticity is also being incorporated. So, we have h max minus epsilon p max by s. So, this is the contact depth which is achieved after the indentation when material has undergone a maximum depth of h max it is a compliance of s and the overall h which is been also taken from here is basically d p by d s. So, that part is being incorporated here with respect to strain. So, we can see the total h c is given by h max minus epsilon p max by s. So, that is the criteria for finding the h c. So, we can see then the by the load indentation depth we can we can we can see what is happening that we are we are first taking the indenting impressing it on the material it is achieving a achieving a maximum depth at a maximum load then while it is recovering it recovers it is also recovering both elastic as well as plastic recovery. So, it some part of it has been recovered that is given by h f that comprises of both the elastic as well as the plastic part. So, the elastic part is given by h c. So, we also to reduce that particular part from the actual contact depth. So, h c will be equal to h max minus epsilon d p by d s or p by s. So, that part is basically being given out here. So, this is the other formula which we can which we have utilized earlier on. So, nano indentation it is very very essential that we derive the information from p h curve and also what is happening at just at the recovery is that that is elastic portion. The true elastic part is nothing but the part which has just been released. So, this is the real elastic part which is which is basically specific to this p h curve. So, we need to derive information from the p h curve and e or the Young's modulus elastic modulus and the hardness is derived from the Oliver Farr method. So, we first have to have a relationship between the h f and the p. So, we have instantaneous depth h f is the final depth of the contact depth and b and m are the fitting parameters. And through that we saw that we can calculate hardness via p max by a c or the contact or the projected area and the reduced modulus is given by root pi by 2 as is the compliance and then by divided by under root a c. So, that that part we can see we can achieve it out the out by that method and again we can also get the we can get the reduced modulus. Reduced modulus is given by 1 minus nu i square of the indentor divided by the modulus of the indentor which is that for a diamond and then for plus 1 minus nu s square divided by e s that is for the sample. So, this is called reduced modulus because it is comprising the this formulation for indentor as well as that for a sample. So, essentially the samples modulus should be little higher than the e r value. So, that is why we are we call it as a reduced modulus which is being obtained from this particular equation. So, this is the part we can see out here that we have reduced modulus which can be obtained from this particular part. So, s is nothing but the stiffness which is being obtained from this particular equation. So, we can see that p is given by alpha h minus h f to the power m where h is instantaneous indentation penetration depth h f is the final depth p is the load. So, m and alpha are nothing but the fitting parameters as we saw earlier. So, we have p as the applied load alpha and m are the fitting parameters and h max is the final displacement which is given by this particular term and this h c which is actually out here. So, we have a elastic part and then we can identify h c s h max minus epsilon p by s. So, we can see the stiffness part is being in the denominator a load and everything else. So, this is the overall relationship between the load displacement curve for the nano indentation. So, we can see once we apply the load we have a loading part which basically goes on like this and the unloading part. So, this is the part which is instantaneously recovered. So, this is the elastic region and this is the plastic region. So, plastic energy can also be calculated by calculating area under the curve. So, we can see that this area is nothing but energy the plastic energy and this energy is called a elastic. So, we can also compare the we can also compare the energies of this particular these two regions to basically compare two different materials how much elastic energy they can store or how much plastic energy they can store also while keeping the relationship constant. So, for a particular loading or particular dispensation depth we need to have this comparison and just stating it forward calculation of area or the contact areas very typical very critical for this hardness and modulus evaluation. Hardness is given by P maximum by A maximum contact and modulus is given by root pi by 2 P and the stiffness is again given by P dash is nothing but d p by d p by d h and A is the area of contact for this one. So, what is happening is calculating this part is pretty much difficult and that depends on the geometry of the indentor as well. So, for a spherical indentor H of contact is much lesser than the radius of the indentor. So, we can get A contact is equal to 2 approximately 2 pi R H of the contact depth. So, this thing is little bit easier, but for sharp and Burkowitz indentor A contact area comes out to be 24.5 H square. So, we can see that from depth or penetration depth itself we can now calculate the area of contact. So, from depth of contact we can calculate the area of contact also we can see that for a spherical indentor the contact here depends on the radius where in this case where in the case of Vicar or Burkowitz indentor it is independent of the any other entity. So, just by once we know the H of contact we can always calculate area of contact, but these are for a pure or a real or a for a defect free Vicar or Burkowitz indentor, but in reality what happens is the overall geometry of this indentor is not always perfect. So, what happens is we also add some correction terms some higher order correction terms as well in this case for Burkowitz or Burkowitz indentor or even spherical indentor. So, we will see that we are not incorporating it out here, but the calibration always requires that we also add some correction terms along with this particular A contact. So, eventually what we get is a reduced modulus which is which comprises 1 by modulus of the indentor as well as 1 by modulus of the sample. So, this is one more typical part of it because then e sample will be little higher than the modulus value what we are getting finally, because once the response is being obtained it is the response of sample plus that of a indentor. So, we if we are calculating the modulus the kind of resistance which is being offered is by both by the tip as well as by the sample. So, we always need to extract or subtract the contribution which is coming from the tip. So, that that is the reason we also need to subtract this part from the from the reduced modulus. So, what we get is little higher modulus for the sample. So, how does the nano indentation works initially we need to measure the overall contact surface area. So, we have certain equation available for that where psi is the semi apical angle for the for the conical indentor. So, in case of Berkovich indentor we always need to have a have a equivalent conical indentor. So, this is for a conical indentor this value psi. So, for a Berkovich indentor we always have a comparative value of psi for a for a conical indentor. So, depending on the angularity of or the geometry of this indentor Berkovich or Wicker indentor we will always have psi for a comparative conical. So, that it behaves like a conical indentor as well. So, we can see the a depends on the pi h square 10 square psi and through this we can through the depth sensing we can always get the value of e which is basically determined from the unloading part. So, we can see for loading and unloading this is the region which is truly elastic. So, this region is truly elastic. So, from this region we can always identify what is the slope and we can convert it back to the reduced modulus from this particular dependences. And also once we have this indentor we can also generate some half panic cracks which can be made in our radial. So, we can see once we have a indentor it will generate a very dramatic or drastic plastic field around it. So, that part again depends on the geometry of the indentor. So, in case of Berkovich or Wicker indentor this plastic region has certain length certain depth as well or it can all for a brittle material it can directly initiate a crack as well. So, we can either have some sort of a plastic or elastic regime with certain depths which are being already been modeled by certain researchers or it can also generate crack when the material is very brittle. So, it can also help opening the crack as well. So, that is given by this particular force or the load which is required to open up this particular crack. So, we can see this once we are loading it we can unload it loading unloading. So, from that we can always find the relationship or the hardness dependence of the load. So, we can also achieve very marginal difference in the hardness because of the kind of load which we utilize and that is given by the hardness is given by load by the area. And then from that we can always identify what is the a value from the semi apical angle for a conical indentor. So, from the depth we can always find what is the contact area and through that we can always find what is the hardness. And again the modulus can also be obtained once we have this particular relationship available of p versus h minus h max. So, we have p is equal to some fitting parameter h minus h final to the power of say some m. So, through that we can always identify this relationship of d p by d h and once we know a we can always fit in here to find the modulus and this is nothing but a reduced modulus. So, we always need to back calculate the modulus of the sample also. So, this is the overall idea of obtaining the hardness and Young's modulus from the nano indentation. From just to say how what sort of constants we can achieve from a projected we saw that it was 24.5 h square. So, for a Burkowitz projected area we can always go about calculating the psi which is for a Burkowitz tip with the equivalent psi angle is 65.3 for Wickers it is 68. So, depending on the geometry we can always find what is the projection area for a Burkowitz tip which comes out 24.56 h square whereas, for Wickers it comes out to be around 24.504 h square. And for Burkowitz tip as I said it is 65.3 is equivalent semi apical angle which is for a conical indentor and equivalent semi angle for conical indentor is around 70.3 as well. So, semi angle is different than the semi apical angle and this provides a similar projected area to depth ratio as that for a Wicker indentor. So, by utilizing this particular geometry we can also see that we can also utilize this conical angle with respect to alpha. So, we can get alpha of 70.3 as well in this particular case. And just by saying that by measuring the constant displacement mode also say if you have two different materials material a with much higher modulus and much higher hardness and then second material is b which is much lower modulus and very low hardness as well. So, just comparing them we can see that how different of a pH curve we can obtain in these two cases. So, in this case if you are obtaining say just for example, say if you have load off to the order of milli Newton's and displacement to the order of couple of hundreds. So, we can see 100, 200 nanometer and say we can go up to say around 50's of milli Newton's. So, we can see that at say around 200, 250 we can achieve a curve and then it will also revert back. So, what we are seeing in this case is that for a very high load we are getting a very low displacement. So, we have h nanometer, we have a depth of around 150 nanometer or so for a load of 15 milli Newton's. So, in this case we can get very high modulus as well as very high hardness because the maximum depth also is very low in this particular case. So, it is to the order of couple of say around 250 nanometer for a load as high as 50 milli Newton's. So, in this case we can see it is might be true for a ceramic material also because the also because the kind of recovery it is undergoing it is also very elastic in nature. So, we can see that much of the elastic much of the load depth has been recovered. So, it is more like a elastic material. So, that is why we can see much of it has been recovered as well. But in the second case what we can see that even smaller loads say around 10's of milli Newton's can generate very high very high displacement also the recovery also is very very low. So, we can see say the curve is like this. So, curve is like this. So, in this case we are achieving say similar sort of displacement. But in this case we can see it is around 300 or 250 nanometer and around say 10's of 5 to 10's of milli Newton's. So, what we are seeing here is this load and this is the depth nanometer. So, we can see that very low loads are creating this similar sort of displacement. So, in this case we in the first case we had 50 milli Newton's to give us displacement of 250. In the second case 5 to 10 milli Newton's are giving us the displacement of around 250 to 300 nanometer also the recovery is very very low. The displacement is very very high it means the hardness is very very low. In second case the recovery is pretty less it means the slope is very very less. So, in that case slope is little bit higher. So, in that case we can see that E value is pretty low it means the recovery is very very less. So, when slope is high the compliance is pretty high it means the material is lesser elastic. So, that part we can see here. So, we can see that there is much more very very less of elasticity in comparison to that of the first case. So, we can see the difference in the modulus and difference in the hardness. Difference in hardness is arising because of similar loads say in this case similar displacement this for a lower is achieved in lower load in this case it achieved at higher load. So, that is why we can see that in constant displacement mode say in this case we had displacement constant at say around 250 to 300 nanometer. So, for a lower load we can achieve that particular depth in the first case it is at much more higher load. So, that is telling the difference in the pH curve itself what we can achieve for two different materials. So, in first case we have a high modulus we see a very very very lower slope. In second case it is very high slope unloading slope also the recovery is pretty high in the first case in comparison to second case. So, that can tell us about the difference between two different materials. So, fracture toughness measurement can can can be done by this particular technique as well. So, basically we people utilize this launch events and martial equation which is which is basically been earlier done by these three scientist that basically takes care of the crack land that that emanates when the indentation is being done. So, for mainly for brittle ceramics the indentation crack which has been generated is assumed to be around half penny shape and under those conditions system will behave like a center loaded half penny crack from which just stress intensity factor k can be evaluated as P r divided by c to the power 1.5 whereas, P r is the crack opening force and c is the crack line that is being generated. So, crack opening force P r which is produced by the residual stress field that results when the peak indentation load is starts to relax and that is being matched by the fracture toughness of the material itself. And psi is nothing but a semi angle or the semi a semi apical angle of the indenter and h and e are the hardness and modulus of the material. So, through that we can also see that how P r can be related to the modulus and hardness and that is being utilized in calculating the fracture toughness of the material as well. So, through that we can calculate we can calculate what is the crack opening force that is being resulting out from here and then from that we can always calculate what is the fracture toughness which can which is responsible for this crack restriction. So, overall fracture toughness we indent the material a crack is effectively generated and when the crack is being restricted that is the characteristic of a material that for a particular load when the stress intensity factor matches the fracture toughness basically the crack will tend to stop. And by measuring that particular entity we can always find what is the fracture toughness of the material. And for this ceramics we always assume that the crack is half any crack and then we require certain load to stretch it further. And again the value of certain constant which is like alpha which is which is already been determined for say Burkowitz or Wicker indenter by using certain semi empirical equations. And again they depend on the geometry of the indenter also because they induce a different stress field at the tip of the indenter. So, evaluating that also is very very critical for measuring the fracture toughness. So, in this case again there have been some more modifications to decide what is happening exactly in case of a indentation crack. So, we can see for a Burkowitz tip we get a we get an impression and then we also start seeing some riddle crack which can emanate from the corners. And L is the length of the crack from the edge and then C is the length from the center. And utilizing those formulation we can see that L and C they are being utilized in the equation alpha is again the geometrical constant and E and H are the modulus and the hardness of the material. So, fracture toughness can be calculated by using this particular semi empirical equation which utilizes the crack which is emanating from the from the edges and also its length from the center. So, we can always find the fracture toughness of the material by using nano indentation technique while utilizing a force enough to initiate the crack and that crack is being basically being restricted. And again just coming to the Burkowitz indenter the semi apical angle of 65.3 or equivalent angle of 70.3 also is out there. So, we can basically what we can see here is the overall a projection area how is dependent on the semi apical angles that is 65.3 and it comes out to be 24.56. So, just to show a relation what is happening is once we have a indenter kind of projected area which is the projected area on this particular part. So, we can see the overall projected area will come out basically for this particular field. And then we can see what is the overall relation between all of them and then measuring the edge in terms of a what is the width of this particular edge. And from that we can always calculate what is the projected area that is being resulting from this Burkowitz indenter and that comes out to be around 24.56. So, seeing the relations out here we have 10.60 that is l by a by 2 and then we can always have 10.60 in terms of the values and then we can see relationship between l and a and then the projected area is given by a l by 2. So, that results the overall geometry and then we also relate b and h by utilizing this semi apical angle that is h by b. And then we can always relate what is h and then relate it to the and achieve its value. And then finally, find the projection area from that by utilizing h and a and then we can finally, get the overall relationship out here. Again there are many many indenters which are also available like Burkowitz or Kono spherical and they have their own functionality or the own dependence. Why because what is happening is like in Burkowitz tip we have very sharp tip and very sharp tip will induce very high plastic strains in case of a hardening material it can induce cracking very effectively in brittle ceramics as well. So, basically it is very good for measuring modulus and hardness values, but in this case elastic to plastic transition is not clear. Because once we have plastic fields around it and elastic fields around it the transition when it is going from plastic to elastic will be very very unclear or it is not clear in the case of Burkowitz tip, but we can get very nice indents, but for Burkowitz it will be like pyramidal. So, it will be more like this. So, we can see very nice impressions using the Burkowitz tip and it will be very very sharp. So, this can provide us a very good estimation of the elastic and plastic for the hardness and the modulus values that is very good for measurement of the modulus and the hardness values, but sometimes many a times we also need to see what is happening. Pickle it to the material if you want to utilize what is the stress strain relationship then we want to go for a conospherical material which will which actually appears more like this that gives us a precise indication or precise relationship between the depth and the contact. So, we can always estimate what is the stress and strain that is emanating from the tip to with respect to the material. In this case the stress field is so high that it is very hard to predict the relationship between the stress that is being generated via implication of this or impression of this particular type of tip in case of Burkowitz tip. But in conospherical tip it is much more easier and tip geometry is not very sharp also the spherical surface is not always perfect. So, that also is some problem with that. So, we always require some calibration to be done for measurement of the contact area, but this is again very very good for doing for imparting very low stress value on a sample surface. So, like if you have to measure a very soft sample like polymers or biological sample, conospherical tip might be very effective in measuring the modulus or hardness of that particular material. Whereas, sharp Burkowitz indented tip will just start to penetrate those soft materials. So, evaluating soft materials it might be much better to use a conospherical tip, but for little harder materials such as ceramics or metals Burkowitz tip might work much better in comparison to conospherical tips. So, there are certain gives and takes between the kind of geometries which we geometry of indented that we finally, tap. So, this is also very very essential component to be considered. And some cases we also require mapping of the surface like we can have an indentation done. Then we also want to see what is the topography of that particular indent. So, we can utilize either something called atomic force microscopy or we can also utilize the same tip as imaging source by using scanning probe microscope. So, we can use the same tip and we can scan it more like an SEM instead of electron beam here we are just sending the indentor itself to scan the surface. So, in conjunction with atomic force microscope the narrow indentation can be very very essential for mapping the surface. So, we can see where we have indent in the surface and which surface or which phase is resulting that particular mechanical property. Also it also an AFM objective also enables to simultaneously measure a nanometer length scale. Because once we utilizing the scanning probe microscope that might be very hard that might also damage the surface. Whereas, in AFM we can utilize very low forces and image the surfaces as low as 1 nanometer spatial resolution. So, they can provide as very good pictures also of the indent also of the surface. So, a simultaneous nanometer scale type of imaging can be done via using atomic force microscope and those limits can be little coarser when we utilize the scanning probe microscope. At the same time we can also know exactly where the measurement is coming from. If we scan the surface using AFM we can exactly pinpoint which finer micro structural feature is resulting that particular mechanical property. Whether it is coming from a nano grain, coming from a precipitate, coming from an interface or coming from a reinforcement. So, we can exactly pinpoint what the AFM what AFM information is providing we can exactly pinpoint by seeing the indentation where it is coming from. So, that addition of AFM becomes very very important in nano inductor though we are getting the mechanical property using nano inductor. AFM is telling us exactly where it is coming from and also nano inhibition has been utilized in many many different research areas for evaluating mechanical properties of bulk nanostructured components and specifically for MEMS micro electronic mechanical systems or nano electronic mechanical systems, biosensors even tissue engineering. So, we can test the mechanical properties of healthy say healthy tissues or rough cartilages or say tucan beak and so many things of biological entities also to utilize mechanical properties of finding mechanical properties of carbon nanotube based composites. So, once we have a reinforcement of a very fine entity such as carbon nanotube, we want to incorporate what is happening at the interface, what is the role of carbon nanotube in improving the mechanical properties such as modulus or hardness. Those have been widely utilized or illustrated by various researchers. In shape memorialize again we can have some sort of martensitic transformation or twinning that can occur. So, those can also be very well identified by using nano indentation also ultra fast laser machining and material synthesis. So, what sort of changes that might occur that can also be interrupt by the nano indentation. So, we can see there are my ride applications of nano indentation which can be utilized in the exact application of these particular fields and then engineering the basically engineering it in terms of certain applications such as biosensors or even tissue engineering or utilizing this composites for certain applications, names, memes or saying about the quality control by utilizing this particular techniques. So, in summary we can see that instrumented indentation is utilized to determine the mechanical properties which is hardness and modulus by using oliver fur method and depending on the geometry of this particular tip we can implies Burkovich, Wicker or conospherical tip for certain various applications. Mainly the indentation is utilized for evaluating the mechanical properties hardness modulus we can map the surface once we have an attachment of say AFM atomic force microscope. We can also see what is the role of particular phase or particular grain or particular precipitate in dictating the overall mechanical property. We can also read about we can also learn about the stress less relaxation that is occurring or even the creep deformation that is occurring in a material. So, this is a very strong technique to give us provide a certain mechanical property evaluation of these nano materials. Also tip geometry as we saw it can have very strong effect on the substrate and that like Burkovich tip is very good for metal lens ceramics whereas, a conospherical tip might be good for polymers and biological materials for extracting the modulus and hardness. We can also do surface imaging by using AFM or atomic force microscope or even scanning probe microscope and that will tell us exactly where the data is coming from to map the surface. So, we can once we map the surface we can somehow related to the multi-scale hierarchy like if we have nano entities how they club up together how do they combine together to give us a mechanical property at the bulk scale. So, with this I will end my lecture. Thank you.