 Hello and welcome back to this 8th lecture on Microsystems Fabrication by Advanced Manufacturing Processes. Basically, let us just do a quick recap of what we had finished last time. So, we had looked into the USM or ultrasonic machining process. We had also tried to find out some estimation about the material removal rate. We also did an investigate this M. C. Shaw's model for predictions of material removal and did some assumptions in this model where we talked about that the abrasive grain should be treated as spherical and also it should be treated as if there are many number of grains between the tool head and the work piece and also that the indentation created by this grain would produce a crater on the surface and for all practical purposes we should consider the amplitude of motion of the tool head to be constant so on so forth. And so there were a set of assumptions that we had made for predicting the MRR or material removal rate and then we started modeling to somehow to estimate what this MRR value would be and in the process of doing that we arrived at a formulation given here in this slide right here where we were talking about the ultimate yield stress of a material of the work piece sigma w and we correlated this to the average force that the tool the vibrating tool head would give on the grains the amplitude of motion of the tool A and the number of grains or particles making impact per cycle D right here was the grain diameter of the abrasive grains grain diameter h w was the indentation depth of the grain on the work piece and this parameter here known as lambda was basically the ratio of the ultimate flow stresses of the work piece and the tool as we already have seen before in details that you know the ultimate yield stress is really inversely proportional to the indentation depth so we can assume that sigma w by sigma t the ultimate yield stress of work piece to the tool is nothing but the inverse ratio of the depth of indentation of the tool to the work piece okay and this we considered as lambda here which comes into this equation here and so therefore we have ways and means to predict the ultimate yield stress of the work material here let us call it ultimate yield or let us say flow stress of the work piece okay so that is how we have a very well defined relation between the various parameters associated with the force given by the vibrating tool head to the area of the grain which is really interfacing with the surface and the ultimate flow stress of the work material sigma w which is in question now if we really try to see what the z value is the grains per impact number of grains in contact between the work piece and the tool in one impact would be so let us say that if we assume that the number of grains acting is inversely proportional to the square of diameter of each grain which is obvious because supposing there is an area like this on which you have so these many grains right these are the grains on that area and the average diameter of the grain is given by D as we have predicted before so this diameter here right here is D so the amount of occupation of the grain area okay would definitely be a function of the overall area that is coming between the tool and the work piece so this is the tool okay this is the vibrating tool head and the tool is coming down like this and the grains are coming in between the tool and the work piece this is the work piece okay and so the the influence of the diameter of the grain on the effective area of the work piece that can be machined which is showed by this shaded region is obvious okay so therefore if we assume that the number of grains acting let us say these are z numbers in one impact between the tool and the work piece so if that is inversely proportional to the square of the diameter of these grains which also is signified or signifying the sort of area of projection of one grain so it will not really be improper to assume this kind of a relationship so for a given area of the tool fair face z is actually proportional to inversely proportional to the square of the grain diameter and also that would be proportional to the concentration of the grains in the slurry okay so if c is the concentration of the grains in the slurry or the concentration term so more is the concentration more would be the number of z's on the more more would be the value of z the number of particles between the tool and the shaded area here work piece so therefore we can say that z is equal to some constant psi times of c by d square and we can actually substitute the value of z from here to here in this particular equation so the final form of the equation can come out to be square of h w is actually equal to 8 times of average force times of amplitude a times of grain diameter divided by pi times of xi and the flow stress of the material is nothing but the hardness of the material so sigma w and h w are kind of inter convertible so h w times of c times of 1 plus lambda okay and where h w now this is small h w as you know as the depth of indentation of the grain on the work piece surface so therefore h w always becomes equal to the square root of 8 f average a d divided by pi xi h w c 1 plus lambda and if we substitute this value of h w in the equation for q you remember q earlier was actually determined to be proportional to d h w to the power of 3 by 2 times of the value of z times of the frequency nu so q here can become of course we can substitute all these values here so the q finally after substitution of h w and the value of z which is actually square of inverse square of d times c times of xi and nu of course is a frequency we get q is proportional to amplitude to the power of 3 by 4 diameter d to the power of 1 by 4 the average force of the vibrating tool head to the power of 3 by 4 times of concentration to the power of 1 by 4 divided by the flow stress of the material or hardness of the material h w 3 power 3 by 4 times of nu and so the rate of removal is through the direct hammering action of the grains due to the vibrating tool so this actually we can say as q direct or in other words q direct is nothing but the direct hammering action of the vibrating tool head on the grains thus creating a plowing action so as I told you there are two modalities of this material you know removal one is of course the direct hammering action of the vibrating tool head and the other that is not very important or not significant although it is to be considered a model is the impact that a free grain would have on the surface meaning thereby if the gap between the tool and the workpiece is very high and there is a possibility of the abrasive grain to freely flow between the tool head and the workpiece so impact that the tool would give on the abrasive grain would be converted as a sort of kinetic energy of the grain and this kinetic energy would come and impinge on the surface thereby removing the material from the workpiece so that separate mechanism so this q that we have determined now is really the direct hammering action where you are squeezing the grain between the tool head and the workpiece and you are giving a force average average force f average between the grain and on the grain by the tool which is creating a direct plowing action so that is what the first part is let us look at now the second part of the problem and the second part is related to the kinetic energy of the grains so let us actually try to model that part so some grains get reflected through the fast moving tool face also impinge on the workpiece so we can estimate the depth of indentation in that case by looking at the following so let us say this is the tool and this right here is a workpiece and there is a grain there is an abrasive grain which because of the motion imparted by the slurry goes and strikes the tool in this particular direction coming out of it in this direction and we will have to somehow predict what is the maximum reflected velocity so this is the direction of reflection of the grain so the maximum reflected velocity needs to be somehow determined in this particular case okay so that is actually let us say y dot maximum. So as we know that you know the grain velocity here of the abrasive grain the initial velocity by which it is striking the tool is of hardly any significance in comparison to the overall inertial component of the tool because the tool first of all is very very heavy and number 2 is it is also vibrating at a certain velocity or ultrasonic frequency meaning thereby that its velocity is also very very high okay so therefore the velocity the initial velocity of the abrasive grain as it strikes the tool surface here for example in this position a here is not really of great significance and we can say that whatever is the velocity of the tool head is at that particular point of collision at the time of collision would be equal to the velocity of reflection of the particle okay so it is simply imparted there is no specific inertial component associated with the abrasive grain because of its small nature it is few microns as I told you abrasive grains could be in the range of 20 to 25 microns. So let us find out first the operating velocity of the tool head as a function of time so y t as you know because it is a sort of simple harmonic motion imparted by the tool so y t can be written in form of an equation as the amplitude of motion a times of sin 2 pi nu t okay nu is the operating frequency of the tool and a is the amplitude and so therefore that is what the equation of motion of the tool head would be okay so the operating velocity of the tool head would be dependent on this equation of motion here and so operating velocity can be written down as y dot t the first differential of y with respect to time which is equal to a times of twice pi nu times of cosine twice pi nu t okay and as you know here that at time t equal to let us say 0 which signifies probably the mean position of the particular tool where the velocity is supposed to be the maximum so this value y dot t would be y dot maximum okay which is actually equal to a times of twice pi nu okay a is the amplitude of motion nu is the operating frequency and so a twice pi nu is basically the y dot max or the velocity of motion. So now we look into the aspect of the kinetic energy of the particular tool once the maximum velocity of the grain is there okay so the corresponding ke or kinetic energy actually will be equal to the maximum kinetic energy because it is half mv square v is the velocity of motion and v is equal to v max corresponding to the maximum velocity the time when the tool is at the mean position okay so therefore the maximum kinetic energy of the abrasive grain I already explained to you before that it really is nothing but the maximum velocity of the tool the inertial component of its own self of the grain is so small that we do not really treat that in this equation and so therefore the maximum kinetic energy of the abrasive grain is given by the term half mv square right and m here because it is a spherical grain that we are assuming with diameter d we can assume it equal to be the volume of the grain which is 4 by 3 pi r cube d by 2 whole cube okay times of the grain average grain density rho a so this is actually the density of the abrasive material okay there is not really the number of grains per unit volume but it is the density of one grain per unit volume of that material so that basically is the mass component in the motion so it is half m v square and v as you know is 2 pi nu a square when nu is basically the frequency is the amplitude of motion of the particular tool head and then this is a characteristic property of the grain itself so if we really try to solve this round we get a term 1 by 3 cube of pi rho a d cube nu square a square where rho a is the density of the abrasive grain so that is what the maximum kinetic energy in this particular case would be cut so problem here so basically now we want to really find out the amount of energy which is really needed for indentation caused on the surface by a flying free flying abrasive grain that comes and strikes onto the tool surface and impinges onto the workpiece surface as a result of the reflected velocity so assuming that during the indentation caused by such an impinging grain the contact force increases linearly with the indentation depth the Ke max whatever has been imparted onto the grain surface or the free flowing abrasive grain by the tool surface should really be equal to half f i dash max h w dash if you remember the plot here cut so in this graph here let us say we assume that f i dash is linearly varying with respect to the depth of indentation h w dash mind you we are using different subscripts here because you know just to differentiate it from the case of direct impact where f i average and h w were the two subscripts which were used there so this is the linearly varying model meaning thereby that when the force is 0 at the beginning and when the grain has not yet stuck on the surface and then the force slowly increases because the grain gives you know all its momentum all its energy to the surface and also faces the reverse force from the surface and then after a while after the full indentation has been realized the amount of force at that point can be treated as f i dash maximum and then you can assume that the grain slowly releases contact meaning thereby that it flies off the workpiece surface and it goes all the way to force equal to 0 ok so the area under this curve here showed by the shaded area is really the work done the amount of work done because of which the indentation has happened so during indentation an area is actually given here by half f i dash maximum h w dash and so we equate that to the maximum kinetic energy of the grain that has been obtained before so therefore you know we can easily find out so sigma w which is actually equal to the also the hardness of the workpiece ok so these are all flow stresses is related to the maximum force at that instant of point when the indentation had gone maximum so f i dash max per unit area so at that time if we assume that you know the total grain dia which has been projected on to the workpiece surfaces so small is capital D ok and capital D is as you already know twice root of d h w where h w is this depth of indentation so if you remember the first exercise on USM that we had done this modeling that how about a grain with a diameter d impacting on a surface producing a depth h w ok so there was a relationship between this capital D here the projected diameter of the grain on the surface and the grain dia ok so therefore force per unit area that you get out of this equation where f goes to f i max f i dash max the maximum force of the grain on the surface per unit the area at that time which we assume to be pi capital D square by 4 ok so we assume this area to be pi capital D square by 4 in other words you can have this as f i dash max divided by pi d h w so that is what has to be equated to the hardness of the surface or the flow stress of the surface for the condition that the grain would actually produce some deformation on the surface and we already know from the previous equation that this k e max can be related to this f i dash max and we would like to now formulate an equation for that so half f i dash max times of h w dash where f i dash max is the maximum force at maximum indentation h w dash this can be equated equal to this kinetic energy maximum which had come from the last derivation 1 by 3 pi cube rho a square of d u a square ok and therefore also from the equation that you have derived earlier here in this particular instance let us call it equation a here ok from this equation a you already know that f i dash max can be equated equal to h w pi d small h w dash where this is the maximum indentation depth of a freely flowing abrasive grain on the surface also thus if you substitute this in this particular equation for f i dash max we get a formulation half h w pi small d small h w dash square half f i max dash times of h w dash is actually equal to 1 by 3 cube of pi rho a square of d nu a square and that way you can actually have h w dash as the under root of twice rho a abrasive grain density by 3 h w times of pi small d nu a ok. So, comparing this h w dash that you have obtained with the earlier h w that was for case of a you know hammered grain or a direct impacted grain we find out that h w is very very high in comparison to h w dash you can compare both parallely. So, if you may just recall in the earlier case the h w dash came out to be this whole 8 force average a d by pi xi h w c 1 by 1 plus lambda ok. So, it really included a lot of terms magnitude wise this h w dash coming from the direct hammering action is always very very high in comparison to the h w dash that you obtained by the free flowing action of the grain. So, therefore, really the maximum material removal rate we can conclude here. So, the maximum material removal rate is highly dependent on the free flowing sorry the direct hammering action of the grain ok. So, it is dependent on the direct hammering action of the grain. So, it can be concluded that most of the material is really removed by the direct hammering and very less amount of material comes out because of the free flowing impact which is really not relevant to mention here also. And from the earlier relationship we already have seen that the M R R Q is proportional to a 3 by 4 grain diameter small d to the power 1 by 4 average force to the power of 3 by 4 times of concentration to the power of 1 by 4 divided by hardness to the power of hardness of the workpiece to the power of 3 by 4 times of nu where nu is the operating frequency is the amplitude small d is the average small d is the grain diameter this is the average force and h w is the hardness of the workpiece ok. And as I already discussed that the Q because of free flowing grains the M R R because of free flowing grains is negligible. Therefore, this really is the M R R value and therefore we safe to say that M R R is proportional to the d power 1 by 4 where d is the grain diameter unfortunately that is not so, but because in experiments it has been observed that the material removal rate is proportional to the first power of d and not d to the power of 1 by 4 here. So, this was a discrepancy that you know arose from the Shaw's model because of which some explanation needed to be given. So, that somehow this experimental data which comes out to be proportional to d could be easily fit inside the you know the data which has been theoretically derived by the Shaw's model ok. And so, therefore, Shaw actually tried to find out in reality what goes on or what happens. So, the discrepancy was addressed by Shaw finally, by looking at the overall shape of the grain ok. So, Shaw actually looked at the grain shape under a microscope and found out that the grain actually is not a spherical grain, but a sort of flowery structure on the surface something like this. And what really was impacting the workpiece surface was not this overall average spherical grain diameter small d as has been illustrated in many times in the model, but this small diameter here of one such you can say this can be a spherulate d 1. So, essentially this is the diameter which would affect the material removal process and it would in turn indent on the surface. The surface area also would be determined by d 1 and not d. So, he very closely monitored if there exists a correlation between the grain diameter d and this small we can call it the projection diameter of the grain d 1 dash d 1. And what interestingly he found out is that yes there exists a correlation where you know these two things can be very well you know correlated as d 1 the projection diameter being proportional to the square of the average grain diameter small d. Meaning, thereby that if this diameter increases d 1 almost increases as a square of this diameter and therefore, it is safe to assume that d 1 is actually equal to a constant mu times of square of d. And this mu can vary between close to one or somewhere less than one and that way you can have a very nice formulation between d 1 and square of d. So, if you actually use this let us call this equation b in the theory of q and q you already know is proportional to this now it is d 1 h w to the power of 3 by 2 times of z times of mu. Remember this is the volume of material removal and this d 1 h w to the power of 3 by 2 is now indicative of the new projection grain diameter which is actually the diameter causing the indent or the impact on the surface times of the depth of indentation h w which does not remain which remains almost same to the power of 3 by 2. And this is correlated by that formulation square of d is actually equal to 4 times of d 1 h w. So, we are taking the modified diameter of this projection which is actually causing the indentation and trying to find the relationship between this diameter and the overall diameter here the grain diameter d. So, if you put this expression d 1 into you know this particular expression here you get that of course, h w as I already told you for a hammering case hammering grain case can be correlated by this relationship 8 f average times of amplitude a divided by pi z d 1 h w 1 plus lambda. And we already know that z is actually proportional to the concentration and inversely proportional to the square of the grain diameter. So, if we put all these together on the equation for q the q equation becomes equal to cube of d times of h w to the power of 3 by 2 times of psi c by d square nu. One thing which is interesting to observe here is that the z value still is dependent on the average grain diameter value for obvious reasons that the number of particles which are making impact on the surface between let us say a fixed tool area and a fixed work surface is really determined by the area of projection of an average overall grain. And the area of projection of an average overall grain is nothing, but proportional to the square of the diameter the average diameter of the grain not the diameter of the projection projections can be many on a grain surface. So, that is why the z value does not alter the z value is still inversely proportional to the square of the diameter because that is ultimately the determinant of what would be the grain to grain spacing. The average diameter of the grain is the determinant of the grain to grain spacing between the tool surface and the work piece surface assuming a fixed tool area. So, therefore, this expression here becomes conveniently changed and q becomes conveniently proportional to the grain diameter the average grain diameter d which is in consonance with of course, the experimental observation. And therefore, this gives you the total prediction of short theory towards the different parameters involved in the material removal rate of USM process. So, what I am now trying to what I am now I will try to do is basically try to evaluate some of the characteristics typical characteristics of how q will vary with what parameter. So, let us actually write this whole thing down here. So, q as you know now is proportional to d times of the average force f average to the power of 3 by 4 times of amplitude of motion of the tool to the power of 3 by 4 times of c concentration to the power of 1 by 4 divided by the hardness to the power of 3 by 4 1 plus lambda to the power of 3 by 4 times of nu where nu is the average frequency. And thus as you know that q would increase if the grain diameter would increase obviously, because there is a direct proportionality between the two. And if supposing all these other parameters like the force average the amplitude of motion the concentration of the grain and the average frequency if they have increases they would significantly impact the q. So, the q increases because of them and if the hardness of the work pieces more then of course, the q falls down. So, q is inversely proportional to it and also so is true about the hardness ratio. And hardness ratio as you have earlier defined is very well defined as the relationship between what the work piece hardness or flow stresses with respect to the tool hardness or stress. So, if the work piece hardness is more it is obvious that the q or the material removal rate would fall down. So, that is in a nutshell what the predictions of Shaw theory actually show and experimentally they have been many times verified by various people that these trends are actually true. So, we would now like to go ahead and look at some of the experimental trends of different you know aspects of the Shaw theory and how actually and theoretically predicted values would differ above a certain limit of one parameter may be. So, one case is the MRR plot of with respect to the feed force or the average force. So, this actually is a plot between the average feeding force f average as you saw and it is obvious to assume that if f average is more than q is more. So, theoretically predicted trends would look something like this which is represented by this dotted line here as if the f average keeps on increasing and the q should increase, but then what is interesting here is that above a certain limit of the feed force let us say above a certain limit of the average force the there is a depreciation of the material removal rate and the material removal rate comes down after a certain critical feed force and that happens because of a very important effect which practically you know almost always happens into in these USM systems which is also known as the grain crushing effect. So, if the feed force is hired and hired to a value that this f average per unit area of the grain actually equals to the you know the ultimate flow stress of the grain itself abrasive grain itself. So, therefore, there is a possibility that the grain itself would be a concrete broken into pieces and there is a crushing effect. So, the number of active grains which are now valuable at that critical feed force would simply you know go down. So, that because they are themselves getting crushed and therefore, the material is almost always reasonable to assume that because of this crushing effect of the grains etcetera the number of available complete grains which come between let us say the tool head and the workpiece are lessened and so would be the material removal rate and therefore, the actual trend of the material removal rate is shown by this particular illustration. So, this really is a critical force above which the grain crushing would start to take place critical force at which grain crushing effect would be observed. So, that is in a nutshell what would happen to the trend of material removal rate with respect to the feed force. So, there are some other interesting factors to be discussed for example, as I have already pointed out that with frequency if the frequency goes high the material removal rate would go high. So, is visible in this particular trend here of course, you know the actual varies slightly from the theoretical although theoretical shows almost a direct relationship linear relationship with increased frequency, but the actual is slightly different because of reasons associated with the inertia of the slurry and the inertia of so the tool head. So, is the case with amplitude as you may recall that amplitude if it increases here is proportional to 3 by 4. So, therefore, sorry to the a to the power 3 by 4 is proportional to the to the Q MRR material removal rate. So, therefore, any increase in amplitude would also record an increase in the Q value which is true here as you see in one of the cases for a certain frequency let us say new one for you know an increase in the amplitude there is a recorded increase in the material removal rate. And if supposing the new operating frequency keeps on varying between let us say new one to new three when new three is greater than new one you can see that there is a double effect. So, one is the effect because of amplitude and another is an overall increase because the frequency domain in which you are operating and mind you frequency is proportional to the Q MRR here is also increasing. So, as you increase the frequency the overall material removal rate with different you know for different frequencies that the amplitude they would have a linear increase. So, we have already studied this aspect the feed force where you saw that there is a grain crushing effect which is there and some other trends that can be useful are that related to the you know what would happen for example, with increasing amplitude and feed force. And so, this actually is illustrated by this particular figure here. So, with an increasing amplitude if the feed force is higher for every feed force there is a crushing critical limit. For example, if the feed force is at a lower amplitude meaning thereby that the gap between the overall gap between the tool surface and the workpiece surface is lower. So, at a certain critical feed force value here the grain crushing would happen and this would keep on increasing. So, the critical limit of the feed force goes on increasing as you can see here at which grain crushing begins. For example, at a lower amplitude it begins much earlier and at a higher amplitude it begins later and that is probably obvious because the gap in this case between the tool and the workpiece is more. And so, you know it is important to see if the gap is more then the critical feed force which would be needed for having this grain crushing effect would actually be higher because the tool has a higher relaxation time for going from the surface all the way towards this other extremity amplitude of motion is more. So, if you have more relaxation time then there is a possibility of crushing to happen at a higher feed force in comparison to if you have less relaxation time in case of a lower amplitude. Also important is that if the lambda value that is the work hardness to the tool hardness as I had illustrated before is increased there is a reduction in the material removal rate which comes obviously because of this equation here as you know 1 by lambda to the power of 3 by 4 is inversely proportional to the mean material removal rate Q. And therefore, it is good to assume that if lambda increases the material removal rate would fall down and these are some of the relative material removal rates for a frequency of let us say 16.3 kilohertz of the vibrating tool head and amplitude of 12.5 micrometers of the vibrating tool head and a grain size of 100 mesh. So, you can see that for different work materials like more brittle materials glass the material removal rate is very high which effectively means that the work hardness by tool hardness is lower in this particular case. And if it is more ductile in nature as is going slowly on a higher and higher scale you can see the MRR is reducing because of change of material here. So, of course, the hardness and the brittleness both of the work material plays a very dominant role in this process. And therefore, particularly in MEMS applications or micro systems applications when we talk about silicon micro machining or when we talk about glass micro machining and they are very brittle in nature. So, the paradigm is really very high material removal rate which has to be well controlled so that you can actually have a small channel imprinted through a masking technology that will probably show at the end of all this fundamental process analysis of the mechanical kind. So, that is what how these processes would be applied to fabrication of micro systems technology. And so basically there are certain other aspects which I would also like to point out here for example, let us say if we talk about how the variation of material removal rate would be with the mean grain diameter. It is obvious to assume that as the grain diameter increases the material removal rate theoretically should be proportionally increasing as you already have mentioned earlier that q is proportional to the mean grain diameter. But again the important aspect of grain crushing comes here because if the grain is too high in diameter there is a tendency of the tool to crush or start crushing the gains as you can see here. So, crushing of grains and the moment this crushing phenomena happens as you know the MRR goes down. So, the actual value of MRR for a higher diameter grain greater than let us say a critical diameter v 1 here would be not following the theoretical trend it would actually start coming down. And so is true with concentration. So, for example you know if you keep on loading the grains in the slurry at higher and higher concentration for two different materials it has been proposed here let us say for boron carbide with a different hardness and grain hardness and silicon carbide with a different relatively lower grain hardness. You can see that with the increase in the abrasive concentration the MRR kind of plateaus and that is because you can always between the tool and the workpiece. Supposing this is the tool surface and this other is the workpiece surface and there are lot of grains on it. So, you can only pack this area available of the tool to its fullest capacity. For example, if you load more number of grains this density of the grains per unit area of the tool workpiece surface would keep on increasing up to only a certain limited value beyond which any further grains cannot be accommodated. So, even if the concentration is increased beyond that any further you do not see much you know material removal because the amount of grains which are at probably the critical concentration here are fully packed into this area. So, therefore, there is a plateauing action of the MRR with the increase in concentration beyond a certain critical concentration. So, is the case with viscosity a very very important term for the slurry particularly when you talk you already know that at the very beginning I had mentioned that the MRR in a USM is really dependent on how or what the constitution of the slurry would be made up of abrasive particles and a fluid medium. And so, if the viscosity of the slurry is more meaning thereby that the you know interlayer shear between the fluid carrying the particles are more there is a tendency that you know it will have a creepy motion or just like molasses it will move very slow and because of that all the material which comes out essentially because of indentation etcetera would not be easily dissolvable in such a situation. So, the diffusion gradients that need to be established should be very high for the debris material which is formulated because of the indentation and the brittle fracture do not get carried away very easily in that case. So, therefore, with the increasing viscosity as you have seen there is a relative reduction in the material removal rate as can be illustrated from this trend here very very important to know that if the viscosity is higher the removal of the material debris that would happen would be kind of at a lower rate. So, that is in a nutshell what some of the trends operating trends would be another interesting factor is what happens you know for a brittle and a harder material. For example, in this case you can compare two such materials of average surface roughness values in microns between tungsten carbide and glass as you can see here. And with a mean grain diameter increased of course, there would be a critical grain diameter beyond which there would be grain crushing which takes place, but what is important here to see that if the brittle if the surface is more brittle then the surface roughness value which would eventually arrive at would be higher in comparison to a more harder material. For obvious reasons that a brittle material would be more amenable to brittle fracture and greater chunks of pieces or materials would come out and they would form in turn larger craters and because of the larger craters the overall average roughness of the surface would be higher. So, these are some of the dependencies of the various parameters associated with the AJM process and what I would like to next do today we are of course, at the end of the lecture, but we would try to design some USM problems and predictively a certain what is the material removal rate which would emanate from such a design. So, in probably the next class whatever theory we have learned by the NC Shaw's model of material removal where we saw that the prominence of the direct impact or the direct hammering is much more in comparison to the free flowing grains and the way that that removes material. We would like to now design some problems in a manner so that we can estimate the material removal rate. So, you have a ballpark idea of what are the rates that we are talking about and in terms of the specific energy that is needed through this process as opposed to some of the other comparative processes we will try to compare and then of course once we are done with all that designing and the very important aspect of tool design would be taken into picture and finally, we would like to apply these to micro systems fabrication technology. Thank you.