 Good morning. We will continue with the interaction of the blast wave with a structure which happens to be in the path of the blast wave. For instance, we have a body, the blast wave is propagating. We continue with this and today we will finish this portion namely blast wave and its interaction with bodies. To be able to finish it, let us first sort of review what little we have done in this. We told ourselves, well a blast wave is generated when some energy is released at some source. The blast wave as it propagates further, we considered only a spherical blast wave because this is what really happens when the blast wave progresses away from the source. We have the spherical blast wave moving away from this. We told ourselves near the source or rather in the near field of the explosion, well the over pressure that means if we take this particular point, the pressure rise from the ambient value P0 to the value Ps behind the blast front or the shock front which we call as over pressure Ps minus P0 is rather very high. As it progresses further and goes away far from the source of the explosion that is in the far field, the value of the over pressure Ps minus P0 is small and we non dimensionalized it and expressed it as a value of the non dimensional over pressure. We learnt how to calculate this. We calculated it rigorously in the near field. We said well using some other theories it is possible to calculate it and we arrived at the expressions for this. This is point one. We also told ourselves in addition to this, maybe if I consider a point here or a point here or a point over here, what happens when I look at the pressure behind the front and sort of I say well this is the pressure behind the shock front or the blast front or the lead blast and I express it as a function of time. After the explosion starts that means at time t is equal to 0, I have an explosion taking place. It takes some time for the blast wave to come over here. That means the blast wave reaches this particular point after a certain time we called it as the arrival time and what happens till then the pressure was the ambient pressure and then the pressure rises and then the pressure fell off. Because of the momentum it could go to negative values, we said we got positive impulses. We also got negative impulses. This is the second thing we learned. We told ourselves because of this impulse well the thing could topple down or it could cause its own damage because of the blast wind. Therefore, we talked of two types of damages from over pressure which was crushing or compression because you have rapid compression behind the lead shock and also damage due to the impulse for a body which is hit by this. But then mind you we also told ourselves or we were able to derive something that if the blast wave comes normal over here, you get something like you have the pressure now is P s in the side view. The wave reflects back. Let us see how it reflects back. Well it could reflect back like this. It reflects back over here. The reflected pressure behind it was such that you had the reflected pressure minus the pressure behind the lead shock P s divided by the value of P s minus the ambient pressure was of the order of 8. It is not just multiplied by 2. It could be of the order of 8 for a very strong shock and for a weak shock well it was of the order of 2. Which means when a blast wave hits it normally well I could have the pressure at the surface as a quite a high value that is 8 times the value of the over pressure what I got from the instant wave. This was something which we derived for strong shock and also looked at it for the case of the weak shock derived it for weak shock. And then once it was over we also talked of something else namely we told ourselves well the damage is not only caused by the reflected shock which is very high and the impulses but it could also be caused by the missile effects. Missile effects because fragments from the casing fragments from the neighboring could come and hit it with momentum and that could also cause damage. These were the three types of damages which we talked of. Having said all this we got into the last part and what did we do in the last part? We talked in terms of bodies having impedance and we told well if I have a medium which has a certain impedance let us say medium A having impedance ZA. We have medium B having impedance ZB and we have let us say the blast wave which propagates from medium A into medium B well something gets reflected. This is the incident wave maybe something gets reflected here you have reflected and something gets transmitted into the second medium and we found that if the impedance of the second medium was greater than the impedance of the first medium well the transmitted shock goes away. You have reflected shock in other words I have something like an over pressure coming and hitting this particular interface. This over pressure is reflected back as a shock again we were able to relate the reflected wave with respect to the incident wave and we also found out the magnitude of the wave which transmits into the second medium. But for the case in which I had ZA that is the impedance of the first medium being greater than the impedance of the second medium what happened? We found we were able to derive this expression again we told ourselves well an incident shock comes over here let us say it comes over here it gets transmitted with a higher amplitude into the higher pressure into the second medium but what gets reflected is not a compression disturbance or a compression wave but it is something like an expansion wave that means if I have the ambient pressure here the pressure falls and I have an expansion fan which causes this and this we found is the results in something like spalling of the material A. Having seen all these three or four different aspects of the blast wave let us now go ahead and try to conjecture a little bit more about the damages about the interaction and to be able to do so let us first put down on the board a few cases like for instance we say well over pressure could cause an interaction and damage may be impulse could cause may be the the missile effects could cause and the variation of the impedance change could cause some particular expansion which would cause spalling. Therefore let us first put down the effects of over pressure that means PS by P0 mind you we are still looking at the end view that means we are looking at a shock being formed we are standing away from the shock and looking at the shock and we find that the pressure rise across the lead blast wave is PS or the lead shock wave is PS and therefore we take a few damages we say well a glass pane breaks when the over pressure is of the order of 1 kPa. The next type of damage which can say is supposing a blast wave hits our ear the ear drum ruptures when the value of PS minus P0 is equal to 30 kPa. The next one I consider is brick a brick wall brick wall fails when the over pressure is 50 kPa next a man who is may be standing in the path of the blast wave he sort of tumbles sort of a man tumbles or he is knocked down when the pressure is something like 70 kPa and so on we can keep on tabulating I just put one more or one or two more I say the lung human lung gets damaged when the over pressure is something like 210 kPa and if a human being is in the path like in the path of the blast wave and if the over pressure is of the order of 700 kPa that is seven atmospheres well it is fatal he just dies. Therefore we say these are the things which we can put based on observations of over pressure and let us take a look and why these things should happen and maybe make some recommendation this is what I do today. Having said that let us take the example of glass pane breaking we put this figure on the board again we say well I have a glass pane glass pane is normally thin well I have a glass pane a thin glass pane over here I have air through which may be the blast wave propagates hits the particular glass pane and well the wave gets propagated out let me put the thickness of the glass pane in yellow color this is your glass pane over here well we tell ourselves if the value of the reflected shock behind the pane is about one kPa well it breaks therefore I tell myself well if I consider ps minus p0 divided by let us say p0 well ps minus p0 should be half because I am talking of reflected over here I am considering the value of ps minus p0 is only one kPa or one hundred of an atmosphere and therefore we say well it is 0.05 is sufficient to break my glass pane this is as per the values of the over pressure. Now therefore we tell ourselves well the type of magnitude of the over pressure well if this is the magnitude let us say with respect to the height this is the pressure therefore if I have a small value of over pressure of the order of 0.05 well this is the type of shock wave I have the impulse behind I have the negative impulse here I have i plus over here I have i minus over here this is the type which the blast wave is coming and hitting the particular glass pane. Now let us say for air we looked at the values the value of the impedance for air we said is around 400 to 420 or 440 let us say it is 420 Newton second by meter cube for the glass pane we had a value and the value was around 17 or so into 10 to the power 6 Newton second per meter cube the value for air again was something like we say 420 and therefore what do what do we expect to happen with the glass pane when it is hit by a very small pressure spike of the order of 0.05 that means the over pressure is only 0.05 kilo Pascal that is it hits it I have reflected wave I have 1 kilo Pascal which hits it therefore what is going to happen well on the left side of the boundary I have this which hits it well the impedance over here is less than the impedance over here that means z of the air is very much smaller than the impedance of the glass over here well it gets reflected then as a pressure wave and well if I calculate the magnitude well the magnitude which will be about the same because I find that the magnitude of the z b is very much higher therefore this does not play a role and with the result the some pressure wave gets transmitted into the glass well the amplitude is now twice because you will recall we had the expression p reflected is equal to z b in the second medium minus in the first medium divided by impedance in the second medium plus the impedance in the first medium is a reflected and what was transmitted was equal to p in b which is transmitted is equal to this is the magnitude of the pulse what is coming this is the magnitude of the pulse which is going through was equal to 2 times z b into z b plus z over here therefore with the result the amplitude is higher and we also told ourselves well the glass pane is something which is not very very thick it is quite thin and therefore the thickness of the glass pane being small well this wave does not decay in the small length and when it reflects back over here the ambience over here is again air I have from a higher impedance it goes to a lower impedance value and therefore this wave gets reflected as a reflected wave and therefore you have particles behind the incident wave being being a being in a state of expansion and therefore it is pulled over here here the particles are pulled in this particular direction and therefore the glass material is pulled under tension it fails and therefore it disintegrates or breaks and therefore you find well the glass sort of develops cracks and it breaks but what happens it takes some time for the glass to break and mind you what is transmitted out of the glass is only a very small portion because what gets transmitted out of the glass in this case this becomes the first medium a this becomes the second medium b and what gets transmitted is a very small portion z b being very much smaller than than a than z a what is going to happen the amount transmitted is very small because this is a high value this is a small value very small magnitude of pressures pi gets transmitted into the medium here I have spalling or the glass which disintegrates and breaks but then what is going to happen it takes some time for the crack to develop fully and break by then you have the impulse we told ourselves the over pressure is small and what is going to happen let us plot it out the over pressure you have this is the ambient pressure your pressure rises it comes like this well it goes like this you know since the over pressures are small maybe the breaking can happen in the far field in the far field these things have long impulses and by the time it breaks well the negative impulse could come and if the negative impulse comes the broken glass will sort of fall back over here that means the broken glass falls towards the explosion rather than it falls away from the explosion well this is the type of failure we express we we expect in the case of a glass pin and this is what is observed the glass pin falls back well the over pressures are small in the far field we could have the the breakage of window pins taking place you know we talked of window pins breaking we said in the texas city disaster something over a distance of something like a few hundred kilometers the glass pins of the buildings got shattered and therefore this is one type of failure we can talk of let us come to the next type of failure where we go get back to this maybe we take a look at the eardrum rupture well we told ourselves well you know when we look at our eardrum you know that the eardrum you know we hear all frequencies that means we are talking of the frequency response being quite high for the eardrum if we talk in terms of the frequency response we say well eardrum has a very high frequency response or the time over which it responds to any sound is very small and now if we consider a wave that means we consider an over pressure it is it is coming over here behind the over pressure there is an expansion it goes over here this is the way the wave travels as it moves forward the next signature of this would be maybe at the next instant of time it comes over here the amplitude p s minus p not as decreased well it will go like this and so on it comes like this it goes like this well the the time keeps increasing maybe I should have drawn the second one to be an increasing line because as distance increases the equivalent time increases this was the Kranz Hopkinson scaling law and in this case well it increases still further it comes like this and therefore what is going to happen if this magnitude of the over pressure is of the order of 15 because we told ourselves p minus p not over pressure that is behind a reflected wave that is the total pressure is 30 behind this p s minus p not it is about 15 the air ruptures and why does the air rupture let us again look at the mechanism and then look at the time response the eardrum is a very very thin diaphragm the properties of the eardrum are something like a muscle we can take the value to be 1.6 into 10 to the power of 6 Newton second by meter cube well for air we said it is around 420 or 430 let us say 420 Newton second by by meter cube over here we also find that the value from air the blast wave comes into the eardrum it gets back into the air subsequently the air over here the value is again 420 and what is going to happen well the the blast wave comes over here it meets a higher impedance it gets reflected this is the incident wave this is the reflected wave what gets transmitted is from the from the eardrum it gets transmitted into there may be a small amount gets transmitted but what really happens here within the fabric of the drum you have a wave and behind the blast wave you have expansion it gets reflected over here as an expansion and the eardrum ruptures because of the tensile loading that means if I were to plot if I were to thicken this I have to just take a small part here I say it is the thickness well I have expansion wave behind the shock wave I have expansion wave traveling in this therefore I have this the the particles pull each other and therefore the eardrum ruptures and this is how an eardrum ruptures but then we tell ourselves we also know that the eardrum is a high response system and therefore the movement I have a p s by p not of the order of let us say 15 kilo Pascal that is the total value required is 30 kilo Pascal that means even before the entire impulse is there just the movement the over pressure is there the because of the high frequency response of the fabric of the eardrum well the eardrum breaks we will take a look at the response times a little later in this class but we tell ourselves well an eardrum ruptures from the tensile failure because of the air air and the drum between the type of over pressure is required is 15 from the value of the value of p s minus p not behind the blast wave because the total value of reflected value is twice because the value of 30 kPa corresponds to the far field which is not very very strong well we looked at the eardrum ruptures well if the pressure spike is much larger the reflected shock wave could be not be twice but it could be three times four times up to a value of eight well these are the two cases let us let us look at the third example the third example we take is maybe a failure of a brick wall let us consider this because you know we do use brick walls for protecting structures we have houses in which we have bricks and all that let us take a look at the failure mechanism of let us say a brick wall it could be similar to this except that maybe a brick if it is placed horizontally maybe I have several of these bricks being placed well the length of a brick is around nine inches therefore it could be something like 0.23 meters and this is the brick wall over here we place the second brick over here mind you let us assume that the joint between the two bricks is same as the brick wall maybe it cemented together with same properties let us assume that a blast wave comes and hits it that means I have the incident blast wave and let us presume that we are hitting it we know that maybe the total reflected pressure here if it is of the order of 50 we say it fails let us assume that I have an incident wave whose over pressure that means p0 over here ps over here ps minus p0 is equal to 50 kPa which comes and hits the particular brick wall well the impedance of air ahead of it we take a value which is around 420 we take impedance of air which is let us say medium a before the brick wall is equal to let us say it is 440 let the unit be Newton second by meter cube the brick wall has an impedance and the impedance of the brick wall is 7.4 into 10 to the power 6 Newton second by meter cube and then what is going to happen since the impedance here is very much higher than this what is happening the initial over pressure of 50 kPa gets reflected as an over pressure of 50 kPa that means this is incident this is reflected depending on the magnitude of this well it is not very weak it is not maybe it gets the reflected pressure could be two times or slightly higher than two times over here and then what is going to happen well if I work out using impedances since I use acoustic impedances and we said impedance is equal to the density of the medium into the speed of sound in the medium well I consider the case of weak shock waves well it is just reflected with twice the value that means I have the pressure over here reflected pressure is equal to 50 into 2 that is equal to 100 kPa and this comes out from the expression what we wrote earlier we said well the value of the reflected pressure is equal to zB plus zA into zB minus zA and if we put down the values we have zB is equal to we had 7.4 into 10 to the power 6 minus you have something like 440 that is 0.44 into 10 to the power 3 divided by you had zB 7.4 into 10 to the power 6 plus 0.44 into 10 to the power 3 well this is very much lower than this and therefore this comes out to be equal to 1 and the reflected wave has the same magnitude as the incident wave around the same magnitude. Now we want to find out how much is transmitted into the medium well this is transmitted into this medium this is P into the brick wall B and therefore what is being reflected let us put it down PB prime is equal to we derived this expression 2 times zB divided by zB minus zA well the same thing the value is 7.4 into 2 into 7.4 into 10 to the power 6 divided by 7.4 into 10 to the power 6 0.44 into 10 to the power 6 is almost like zB which is equal to 7.4 around 7.399 into 10 to the power 6 which is of the order of 2 that means I have a pressure amplitude that means let us plot it in blue well this is the amplitude and higher twice this value that means 100 kPa gets you have 50 kPa coming over here therefore the value of the reflected one behind it is equal to 100 kPa and what gets transmitted is equal to 100 kPa gets transmitted into the brick wall well this goes down let us presume for the present that the distance of over the distance of 2 3 meters the amplitude does not get decayed it starts with this in practice over this distance the PB will keep decreasing because as distance moves what did we say as rs by r naught increases the over pressure decreases but in this case let us assume it is a small value and therefore what comes at this interface between the brick wall B and air over here what happens is you have the same value of PB coming over here and when the same value of PB comes over here what is going to happen well it meets lower density a lower impedance air over here which is 440 into Newton second by meter cube and therefore what is going to happen you know the magnitude which gets reflected now has a value you have the value I think there is a sign mistake over here we said reflected over here it should have been ZB minus ZA is a reflected value divided by ZB plus ZA over here and here also it should have been ZB plus ZA and therefore what is going to happen now you have you have the value of B which corresponds to air being very much smaller than the value corresponding to this therefore the reflected wave Pr prime is equal to if I take the value that is going to be 0.44 into 10 to the power 3 minus I have 7.46 into 10 to the power 6 divided by I have ZB plus ZA which was same as the value of around 7.4 into 10 to the power 6 and therefore I find that the similar magnitude gets reflected over here but as a as a rare fraction or an expansion wave rather what comes over here is a strong wave over here it comes over here with an amplitude of 100 and what gets reflected is as an expansion that means instead of compressing it expands over here and since this behind this expansion wave particles move like this here the particles move well the brick collapses over here because brick cannot take tension but more importantly what is getting transmitted what gets transmitted into the air over here if I were to take a look at what gets transmitted let us write it in pink color chalk over here what gets transmitted is equal to what gets transmitted is two times into ZB divided by ZB plus ZA in this case B is the second medium which is 440 that is 2 into 440 divided by this this value is around 7.4 into 10 to the power 6 again is what is and we already know that is 2 ZB we have the value which comes here is something like we said 50 plus 50 is 100 100 comes over here therefore it is equal to 100 over here and therefore you find this is equal to 0.44 into 10 to the power 3 that means it is very small therefore what is going to happen if I have a brick structure housing brick structure and if a blast wave comes and hits it and people are inside the house you know the the transmitted blast wave into the house is a very very small part of what is getting into the brick but however we find that the brick collapses when the when the over pressure which hits it is greater than around 50 and therefore the brick crumbles and this brick can fall on this on the people and cause damage to them but it is not the blast wave which is going to cause damage and therefore when a building falls on on the subjects over here and still causes damage we say that the damage is tertiary that means it is not a direct damage from blast wave but it is rather the collapsed building which causes causes the disaster therefore we tell well the well the type of damages which are caused could either be primary and primary is from the blast wave directly hitting something which causes this which causes the damage it could be secondary in the case of secondary it could be due to missile effects primary could be from over pressure and impulses secondary could be from missiles and fragments which have the momentum and it is the momentum which goes and knocks down a person or or causes injury to the person and the third is tertiary which is due to the effect of a collapsed building on the human being therefore but how do we make a wall which is which is explosion proof or such that we save some people well that tensile strength of the brick or the concrete over here which is used instead of the brick must be able to take tensile failure and therefore we make better concrete systems maybe instead of brick we use concrete and in concrete we put some slag such that the tensile property of the concrete is improved and we if we can now think in terms of a situation wherein we make the wall of the building little thicker well let us put that figure down in fact we have a thick concrete wall and then if a blast wave comes and let us say something gets transmitted into the blast wave we have the incident values P i prime what is get transmitted is let us say twice the value but still it is quite high if my thickness of the wall is something like very large by the time the blast wave comes over here the reflected value depends on the value which strikes this interface between the wall and the air over here and if this value is now decreased from the value of the large value over here because of the larger thickness well the expansion will be very weak and it will not be able to spoil or cause damage to the concrete over here therefore whenever we use a concrete wall for protecting we not only make the concrete to be higher to have higher strength and tension but we also make the wall to be thicker and such type of construction when we when we use for protecting human beings or maybe from blast is known as a blast wall we use blast walls not only for human beings but to protect some equipments maybe we use it in places where we expect a blast wave and this is used as a protection structure well this is about protecting people and human beings and how to go about this maybe I could also think in terms of a different situation maybe I could have steel we said steel has a higher value of the impedance the value of impedance for steel was around 46 into 10 to the power 6 Newton second by meter cube and you know if I can use steel maybe a smaller thickness because steel can take some amount of tension compared to concrete I can also use dual structures wherein I use maybe steel with some other material over here composite structures such that I I I just keep reducing the values of the over pressures and it is possible to use such structures to mitigate the influence of the blast wave mind you see in these things we presume that the structure of the brick wall or the structure of these composite things are static over here when the blast wave hits it they do not relax but if the blast wave maybe if the boundary conditions by which these structures are held allows the way allows the wall to relax maybe I use something like a structure here which moves in this particular direction well in that case my reflected my transmitted wave will be much lower because I have some structure which is moving and when it moves well it it takes it takes it relaxes the type of the it reduces the strength of the wave which hits the particular wall therefore the boundary conditions are important but in this case when I have a wall which is rigid well I can presume that maybe the blast wave comes it I have the reflected wave and I have the transmitted wave and this is what what happens in practice if you want to protect some vehicles maybe I would like the ends if I if I if I want to relax it well the ends the depending on the end conditions the the structure could move if it moves well it it absorbs some amount of shock over here and the transmitted shock will be even lower having said that let us let us go to some other examples let us go to the last example we take and then we will generalize it how about a human being who is struck by a blast wave and because what happens is maybe at 70 kPa we said well a person who is maybe standing at some place gets knocked down what is this knocking down let us take a look at this we take the example let us say there is somebody standing over here well an over pressure followed by an impulse comes and hits him because of this what happens well he he he is because of the over pressure he is just forced he is displaced because of the impulse what is going to happen I give him a momentum I give him a velocity that means he gets some velocity and therefore the when he moves slightly there is some drag force because of the wind effect he is dragged behind and because of the drag force what is the drag force d you have rho v square divided by 2 the velocity with which he is dragged forward the sides are he is pushed in this particular direction into the drag coefficient over here because of that maybe he he is displaced over here and ultimately maybe he falls on the ground something like this that means he is toppled down and why does he get toppled down because of the impulse that is the positive impulse which provides him with a well with a velocity in addition what happens is the over pressure sort of displaces him whereas the impulse maybe knocks him down we said an over pressure of 70 kPa is something which knocks him down well it pushes him and the impulse behind it knocks him down therefore maybe in this case I should consider the failure or the or the type of destruction or the knocking down is more due to impulse than it is due to the over pressure the over pressure will only displace him like like like for instance let us consider a man who is stout who is heavy who is hit by a wave well he can withstand the over pressure but as he is dragged down by the by by the velocity by the by the blast wind well he topples down therefore let us try to put this in in a in a better scenario I I show it in a particular slide you know I show here an object over here and this object is being hit by a by a blast wave over here it comes inward it hits him over here when it hits this particular irregular sized object over here well it gets reflected well the reflected pressures are very much higher than the incident values depending on the strength of the initial blast wave what we get and well the reflected blast wave moves outward in this particular direction because the object is not very very much like a straight line or something which I have assumed over here maybe I have the initial spherical blast wave coming over here the characteristic length here is small and therefore I represent it by a straight line you know because of the reflection you know you have uneven pressure distribution on the surface and therefore you you have the loading on the on the on the particular body over here is not very even in addition to this what is going to happen the blast wave when it comes in these particular portions it gets expanded over here the blast weekends and far away well here it could still be stronger near the body it weakens and it progresses over here therefore you have higher pressure here lower pressure here therefore maybe there is some wind over here therefore what is happening as the wave moves forward maybe you are from the higher pressure region towards the lower pressure region some wind blowing well it creates an eddy or a or a vortex over here I have a vortex over here and the blast wave ultimately which moves through the air gets here there will be some element of transmission of the blast wave through this we have not put this in this figure but what I want to show is well I have the velocities which are induced over the surfaces and they cause the drag and because of the drag well the body gets knocked down and this is the thing but what is more important is well the blast the over pressures over here need not really be even and you find that the body is subjected to over pressures here which are not of the same magnitude because of the distances here the incident wave is a little weaker than the wave over here here the incident wave is stronger because it is nearer over here and therefore we find the pressure distribution over the body is itself not not uniform here you have expansion taking place and because of this you have a force over here which in addition to the wind causes the body to be knocked down therefore what is it we can now summarize the whole thing by telling the following well we looked at three or four examples and based on these examples we find well both the over pressure and the impulse that is the wind effects are important but can we say a little bit more we also talked in terms of the eardrum having a very high response that is the response of the human here is so small that it is a very small number and therefore we said well it could be due to the over pressure itself impulse will not play a role because even if I have a short impulse well it the impulse is over a period of time much before this time well the ear ruptures because it is thin and the reflected expansion waves at the second interface causes the failure therefore we know we have to somehow relate the time of the impulse that means we are talking of positive impulse t plus we are also talking in impulse over here t plus with respect to the characteristic time that is the frequency response of the body which I can say is tau I can tell well the eardrum has a characteristic time tau which is a very small number whereas the impulse has a characteristic duration let us say t plus or some value of t over here how do I relate these two is the question therefore to be able to understand this let us take one small example of let us say a structure or some particular body which is resting somewhere and we all know well it is possible to represent a body let us say of mass m and you know there it is resting against some place therefore I can also model it in terms of a spring mass system well I say equivalently I have a spring of stiffness k and it is held may be over here and what is going to happen well it is hit by a blast load therefore I have something like an over pressure and behind it I have the impulse coming over here and what is happening is I have the pressure initially is p0 rises to ps then it comes to p0 again and it reaches the value of p0 again this is the positive impulse this is the negative impulse therefore because of this at different instance of time I have different pressures which the body forces or rather I have a force as a function of time or rather I can tell myself the force which the body experiences is when the lead shock may be at this particular time when the lead shock comes over here well the force reaches the maximum and thereafter the force decreases and therefore this particular characteristic can put as in terms of force being varying with respect to time. Now therefore I have a periodic or a time dependent force striking a body of mass m which has some in it sort of a spring over here some value of holding the body is here therefore I can represent this free body diagram of mass as m I can say well if I consider spring of stiffness k then I have k into x over here which is the resisting force I have the body force over here ft I have the force that is the force on the free body diagram it is m into x dot over here this is that let us say the direction of x over here this is the acceleration and therefore I can write my equation for the body as equal to mx2 dot plus kx is equal to the time varying force over here and now if I look at the frequency response of this particular body well the frequency is equal to under root k by m stiffness by the mass of the body well stiffness as units of let us say Newton per meter body has units of so much kilogram m as units of kilogram therefore you have Newton per meter kilogram over here this is equal to kilogram meter per second square this is this is therefore equal to I have kilogram meter per second square is Newton I have meter over here therefore this becomes therefore the unit is one over second and this is the frequency so much one over second and if I look at this characteristic frequency the frequency is very high well the the the characteristic time of response of the body is small and therefore if I have a body which which has a very small response well the over pressure itself is sufficient to cause the damage whereas if the characteristic time t of the body is large what do we mean by large large as compared to the or significant as compared to t plus well the impulse will cause a damage and maybe what is going to happen is the average value of the pressure is going to what is going to cause the damage plus the impulse is going to cause the damage can I therefore put these things together in some form of a figure such that I illustrate where impulse is going to cause a damage where over pressure is going to cause a damage let let us try to see whether I can do that what I do is I have a figure in which I put over pressure on the y axis that means I put p s minus p not over here I put impulse on the x axis well you know for some bodies we find that well impulse is really not a matter of concern it is only the over pressure which causes the damage because the response is small therefore I say well this could be one one one case over here and this particular region is the region wherein only over pressure type of causes the damage I could also have the other condition wherein I could have something like a large value of impulse which causes the damage maybe even at smaller over pressure I could have large values of impulse which causes this therefore this becomes my second zone over here therefore what is it you know we are talking in terms of a situation wherein maybe this is the limiting asymptote corresponding to impulse therefore I say this is the impulse asymptote well here it corresponds to the minimum threshold value of over pressure or over pressure type of an asymptote over here and what is going to happen this happens in this region we tell that the characteristic time of the body tau is of the order or very much greater than the impulse time that is t plus or equivalently the impulse time in this case t is very much smaller and in the region wherein you have t of the order of magnitude that means the characteristic time of the impulse and the characteristic time of response of the body are the same well it could be the dynamic region in which case I have to do the dynamic analysis that is the dynamic range over here and therefore if I were to put the value of let us say an iso damage line for both over pressure and impulse what is it I will get I will get the value as something like this coming over here and coming over here and this becomes something like an iso damage curve therefore we could talk in terms of maybe impulse induced damages when the frequency is large or the characteristic time of the body is small of response of the body is small we talk in terms of these damages here when the time constant of the body is quite large compared to the impulse times and when both are of the same order maybe we should do some dynamic analysis or this is the dynamic range of this and to do this is quite involved maybe we will have to look at the frequency response of the bodies and put together that over pressure and impulse together well this is how we look at damages and let us continue with this let us take a look at what are the type of impulses which we get on the body and the ambient and how they amplify each other we will consider one small example now let us revisit the example of crater which we talked earlier we talked the example of a crater what happened in the case of a crater let us put it together you know in the case of a crater we have a blast wave which comes towards the earth well the blast wave hits the earth gets reflected goes up and we also talked in terms of additional waves which are being formed let us take a look at it well this is the earth over here well a blast wave originates from let us say air over here blast wave comes over here strikes over here then what happens something gets reflected let us take a let us plot the value of the reflected values well the reflected wave comes over here now we told ourselves well the reflected pressures are higher and therefore and the these reflected waves are traveling in a medium which is already heated by the incident wave plus if there is an explosion the other heat from the explosion is also going to heat the medium with the result the reflected waves travel faster and they are even stronger and with the result when these reflected waves impinge on the earth over here if the angle is quite small well I get the reflected waves over here but as this angle becomes shallow what happens is I am not able to maintain you will recall we did this in our class we told well if the angle is such that it is not we are not able to meet the conditions wherein the velocity on the ground has to be parallel to the ground well the the incident wave that is the reflected wave which strikes it gets spurted up from the surface and what I form is something like a mark stem shock and then I have a shock like this therefore I have a mark stem shock a reflected shock an incident shock and of course as all of us know we have the transmitted shock into the ground which forms the crater now we will not look at the crater formation we have looked at the what is it he is going to see what is the type of shock pattern or the impulse pattern he is going to see well let us sketch it out well we tell ourselves I am looking at P minus P naught divided by P naught this is the pressure to which he is going to be subjected to well at this time after the blast wave hits the ground after some time he sees only the ambient pressure and after a particular time what does he see he sees well the wave is going to go and hit him he sees this particular mark stem shock or the reflected shock which comes and reflected shock from the reflection which comes and hits him and then after some time well the second shock comes and hits him and may be so on the third shock goes and hits him and therefore the type of impulse what he sees is not going to be a single spike which came down like this but a series of spikes and in general the impulses because of the ground reflection the other shocks which are there cause a series of spikes in the impulse pattern and we seldom get an impulse pattern which is really something like this but we do get spikes over here because of the additional shocks which come sometimes these spikes could be even larger than the initial value it all depends on the ground conditions on the type of reflections what you have but we must also remember when you have an incident reflected pressure over here at the surface you could have the seismic wave and this wave is also traveling along the surface but what causes the damage is the impulse from this particular one well these are about the different ways of looking at it just to wrap up the subject what I do is in the last one let us take a look you know these multiple shocks in an impulse are not only due to let us say due to when the blast wave hits the earth it could also be let us consider the last example wherein I have a sphere and in the sphere let us say this is the sphere gets exploded that means the initial volume of the sphere let us say we put it back in terms of what we learnt we say that the radius of the sphere is Re a sphere which explodes and when the sphere explodes what happens is a blast wave takes off over here now this is the high pressure gases which explode the cylinder it could be hot it could be cold but all the same it is high pressure and then the interface of the high pressure will also follow it will also expand out like this the high pressure gases expand over here and when I create an explosion that is a blast wave in this particular direction well here the gases are at high pressure and what is going to happen a wave is going to travel back that means an expansion wave is going to travel into the body into the medium which has just exploded and because of this interface over here this is the symmetry line well this reflection wave this expansion wave which gets reflected is again coming over here and I have a reflection wave going over here and now what is happening the dotted line red may be I will show it like this in this region well this is a high pressure gas region which is gradually expanding out this is the ambient pressure over here I have an interface here well the impedance of this is lower than the impedance of this therefore the rarefaction or the expansion wave you know when it comes here well it gets reflected as a shock wave for the same reason you know when a shock wave comes it gets reflected as an expansion when a rarefaction wave comes it gets reflected as a shock wave let us show it in yellow thumb that means it gets into the medium as a shock wave over here well it gets again reflected as a shock wave and therefore what is going to happen it progresses into the medium over here it goes into this medium well over here I get a rarefaction wave or a wave which is an expansion wave and therefore when this comes as an expansion wave over here well you know it gets reflected as a shock wave and therefore I have a wave like this and this wave comes forward and therefore I have a series of shock waves one shock wave two shock wave three and therefore if a person is standing over here or maybe at some particular time he is over here he sees the first wave then he sees the second wave and then he sees the third wave at the nature of the impulse what we see is this is the positive impulse and this is the negative impulse therefore the impulse need not be something which is smooth like what we said it could contain a series of shocks one after the other and this happens in all explosions the interaction between a blast wave and a body placed on the path we looked at the mechanisms of failure and maybe with this we will stop our discussions on blast wave we will revisit it again when we look at ideal and non-ideal explosions and also look at how to quantify damages we will try to quantify damages in term statistically because we also recognize that we told ourselves in this class that may be the eardrum ruptures at let us say a pressure of 30 kPa may be a small child who is more sensitive the eardrum of the child may rupture quite early because still very tender whereas some an older person will be able to a strong person might be able to withstand a pressure of 50 kPa therefore to some extent the damages are statistical in nature and therefore later on in the course we will look at damages through a statistical procedure well then thank you and in the next class we will try to take a look at energy released from the explosion in explosions thank you and announcement please in the last 12 lectures we have covered the basics of what constitutes an explosion and the modeling of blast waves I give in the downloads of this video course additional reading material namely references relevant to these portions a small set of homework problem is also given to help you work with the basics which we covered in lectures 1 to 12.