 Now, I will start quantifying the soil matrix or soil aggregate. So, this is a new topic soil aggregate. Basically, this defines the description of the assembly of particles and from this point onwards, the quantification of the soil matrix starts. We also call this as phase relations. So, simple thing is that if you have a soil mass, I am sure you must have realized that now I have switched over from discrete particle of the soil to soil mass. As a first time, I am using the word soil mass or the matrix of the soil until now I have been dealing with mostly discrete particles grains. Now, I am talking about the soil mass. So, this soil mass contains all the constituents which we have talked about, alright, different constituents. Now, suppose if this soil mass has the volume of V and weight W, capital W. Now, this can be depicted as a system of three phases and these three phases are, it is an assumption simplification. So, this is the air phase, this is the water phase for the sake of the simplicity for third year students of soil mechanics, geotechnical engineering. I am talking about simple fluid phases, water and air. Tomorrow, if you want to manipulate it to a multi-phase system, you can do that. Air might be having vapors, fumes of different chemicals and so on. Water could be in semi-frozen form, liquid form, vapor form, of course, which constitutes again the air space, alright. And the solids, you have talked about the air phase, we have talked about the water phase, we have talked about the solid phase. Individual particle of the grain is incompressible, clear? Now, if you do this type of mathematical modeling, this becomes discrete modeling. You are considering each and every particle in the continuum. Continuum is the one which is a continuous system containing several millions of particles of the soils, clear? However, an assemblage of the grains is going to be compressible, clear? So, it is a very tricky material. Sometimes we talk about the incompressible grains, but now the matrix of the soil mass is going to be compressible. Still, we assume that this solid phase is not going to be compressible as compared to water, which is totally incompressible and air, which is going to be compressible at not STP, standard temperature pressure. So, this is going to be, you know, getting compressed if you elevate the pressures. Under STP, we assume that air phase, water phase, solid phase is the way it is. Now, to differentiate these, what we will do is, this is what is known as the three phases of the phase diagram, where the research is. The research is somewhere like if I want to make it a multi-phase system. So, in multi-phase system as I said we can take different components and this becomes a very complicated system. So, if I depict volume of air as VA, volume of water as VW and volume of solids as VS, alright? And the weight associated with air, WA will be tending to 0 or equal to 0. This is okay. We normally do not talk about the weight of the air. So, I will remove this. There is no point in writing this. However, the weight of the water component is WW and the weight of the solids is WS. Solids are basically the skeleton of the soil, minerals. I hope you can realize that capital V would be equal to VA plus VW plus VS. And in this, what will be the air and water combined together? This would be equal to volume of voids plus volume of solids. So, VV becomes volume of voids. Is this part okay? Now comes the question what you are asking. The first attribute of the soil mass is its void ratio. Most of the Indian authors write it void but normally I prefer this as voids ratio because there is no one void. There are several voids present in the system. Now, E is defined as volume of voids divided by volume of solids, alright? Porosity is the derivative of the void ratio and the way the porosity is defined is normally we define this porosity as eta. We have used this symbol ever, eta. So, this is the porosity and this is normally defined as volume of voids divided by total volume. Is this part clear? So, this is what I have to be asking. Is this okay? So, now we have defined the porosity and now you can realize that porosity and the void ratios are interlinked with each other. Compute quickly. What is the relationship between void ratio and void and porosity? Sir, how come? Water volume come under voids volume. Because this is the void space. This whole thing is voids. Voids are the ones in which the water and air is going to sit. The voids are the ones in which the air and water will be sitting in the soil matrix. This is okay. So, when you say voids, this is the total volume of the voids divided by volume of the solids. Now, when I say porosity, this is the volume of the voids divided by the total volume of the system. Is this okay now? Let me defend him. The only way to defend you would be, suppose if I am dealing with only dry soils, water will not be there. The voids will be only of the air. Maybe this is what you wanted to say. This is okay. So, we have created a subclass. Is this correct? Now you are happy. So, you are also correct. I am also correct. But please make sure that the general equations are these. Now, my question is can you relate these two quickly? Do it quickly and find out the relationship between this. F is a function. You are right. Do it from the first principle. So, I will just quickly tell you how to do it. So, your E is Vv upon Vs. This will be equal to Vv upon Vv plus Vs. Is this correct? So, I can write this as 1 over 1 plus Vs upon Vv. Can you modify it? What is Vv upon Vs? E. So, this is equal to E upon 1 plus E. Why? Yeah. So, this is Vv and Vv of Vs upon Vv. So, this is 1 upon E. So, this will be E over 1 plus E. Clear? This is the most fundamental relationship which defines the soil matrix. Now, come out of this discrete systems. Now, we are talking about the soil matrix. So, what we have done is we have defined the void ratio. We have defined the porosity and we have interlinked porosity with the E upon 1 plus E. The second fundamental thing would be saturation. Normally, we define this as S. Those of you who are more into hydraulics would define this like SR. Both are correct. So, sometimes they define it as S or SR. I prefer S to avoid the complications of another R, you know. This is defined as volume of water divided by volume of voids. And remember, all these terms are in percentages or you may convert them into fractions also. So, this could be either in percentages, alright? Or I can use fractions also, both ways it is used. So, this is the saturation. What is saturation? The next term is if I am defining as saturation, this is something known as air content Ac and this is equal to 1 minus S. So, basically this can be written as volume of air divided by volume of voids. The next term in the series is gravimetric moisture content. Normally, this is defined as a small w, again percentage. Can you guess what will be the definition of the moisture content? Vw upon why? You are correct because the volume of water divided by volume of solids is the w. But normally, we do not write like this. We were talking about gravimetric and you brought into it. You got it? So, you get 0. Is this clear? This is a fundamental mistake which many people do. So, please remember, we were talking about the gravimetric analysis or we were talking about the gravimetric things and you have included the volumes is not correct. It will be in the terms of weights. So, weight of water divided by weight of solids. Is this okay? Now to confirm your question, now suppose if I say volumetric moisture content, normally this is defined as theta. 30-year students are not supposed to know these things but I am trying to expose you to these concepts which is equal to porosity multiplied by saturation or sometimes we can also prove that this is moisture content multiplied by gamma D. Is this okay? Now gamma D is the dry unit weight. When I say soil, it is understood that I am talking about the soil mass. We will discuss how to obtain the gamma D value. Let me complete first the series. Then comes the unit weight. Unit weight of the soil mass is normally defined as gamma. We define this as the gamma T which is corresponding to total. Very soon we will realize that there are different types of unit weights. Clear? So, gamma total is the weight of the soil divided by volume of the soil. How it is computed? So, first let me introduce the concepts of the unit weights. These unit weights are number 1 total, number 2 submerged. Any idea how would you compute the submerged unit weight? Please remember this is not the density. These are the unit weights. So, density of the material multiplied by G value. Agreed? Units are kilo Newtons per meter cube. Especially gravity G is gram per cc. So, submerged any idea how would you obtain this? So, let us understand what this concept is. If I take let us say this is the ground level. Water table is somewhere here. This is how the water table is depicted. Alright? If I take a point somewhere here, let us say point number A. There is no direct connection with water table. Depending upon the material, please remember. Depending upon the material, at this point the unit weight could be gamma D provided this layer happens to be sands. I will explain what is the difference. Clear? So, we are just talking about the physical attributes of the material, coarse-grained material. It cannot hold water. Alright? And if there is a water table, we will soon study that there cannot be any capillary reaction and hence water cannot rise in this layer. At this point, the unit weight is going to be gamma dry. However, if I change these sands to the fine sands or clays or compacted clays, the story changes. Simply mugging these equations is not going to help you unless you realize the concepts behind everything. So, I have created several situations out of it. Are you getting this point? So, if I say large grain granular material, water table is not going to influence it ever. But the moment I say fine grain sands, fine grain sands will have a tendency to create capillarity and this water will get lifted up up to a certain depth. Clear? Or if it is clays, very fine particles and the material has a tendency to suck water because of capillary action. Fine? Compacted clays. You are using clays and you compacted them. So, you have created more and more capillary action. We will discuss all these in details. So, now this point is in the dry state. However, if I consider a point B and if water table remains over here, most of the time this point B is under submerged state. Clear? And submerged state is nothing but the total unit weight minus gamma w. Normally, this type of unit weight is depicted as gamma submerged. Please be very, very careful with the birdings of the situation. The material is so notorious at the same time so obedient that it wants to understand what the circumstances are. Clear? Try not to do these type of mistakes. That means you have to read very carefully the type of situation which has been created when something is being asked to you. So, I have created several situations, material property and the way their deposition is occurring. Is this part clear? We will discuss about this much more. These are normal mistakes people do. So, I have defined now submerged unit weight and the third one could be saturated. So, this I will say as gamma sat and the fourth one which I will be using is dry unit weight which I will be defining as gamma dry. So, first of all 4 types of unit weight I am talking about unit weight of the soil mass, gamma total, gamma submerged, gamma saturated, gamma dry. Clear? Now, coming to this point, what I said is this is the ground level. You have the water level. A point A over here is in the dry state as long as the material is not a fine grained material and not compacted material and it is a granular material which cannot hold water for a long time. That means capillarity is not going to be in this region at all. However, as long as the water table remains intact over here, any point which is lying over here is in the submerged weight, submerged condition. And when we talk about the submerged, sometimes people also call this as buoyant, buoyancy you have studied, clear? Is this perfectly enough? So, this is a differentiation. The way the layers are placed, the point where it is and the material property and the complexion effect, fine? Yes, because this is submerged. So, what you have to do is this unit weight is going, see you have to subtract the unit weight of water because this is submerged. So, this is sort of a buoyancy which is acting at this point. So, unit weight of this point is going to be gamma submerged, gamma buoyant, clear? So, either you write gamma B, sometimes they write gamma B also. It is a good idea to follow the symbols which I am using in the class so that there is no confusion between both of us at least, alright? You agree? Is this fine? So, this is either gamma B or gamma submerged, gamma total, gamma saturated, gamma dry. Four types of gamma we will talk about. Take few minutes to assimilate these thoughts in your mind. Yes, this is the volumetric moisture content which is porosity multiplied by saturation. And this is equal to volumetric moisture content multiplied by gamma dry. Theta, there is no formula. Normally, what we do is, theta is directly measured. So, theta, okay, I will tell you. If you are interested in knowing theta, theta would be volume of water present in volume of whites. Now, try to prove it. This will be a good homework for you. Now, I think you are the guy who said clear gravimetric versus volumetric. Now, what we have done, this is the gravimetric thing of the weights and now you are talking in terms of volume. So, look at the volumetric moisture content. This is volume fractions. The moisture content in gravimetric form was the weight fractions, clear? Sir, the formula for the saturation and thetas we have written same. Saturation and? Theta. Theta. Yes, sir, VW account. No, no, Abhi, do one thing. You please do this as a exercise, clear? Sir, like we have defined gravimetric moisture content as WW account WS. So, theta should be also equal to VW account VS, VW account VS. That is what I am saying. So, please remember, the volumetric moisture content is always defined as the volume of water or the moisture which is present in the volume of the whites, clear? So, this is volume of water which is present in volume of whites. What is gravimetry? First of all, in chemistry you have done gravimetric analysis. You weigh everything, clear? So, that is gravimetric weights, volume, volumetric moisture content, clear? So, volumetric moisture content cannot be weighed. What you have to do? You have to measure it. This is the volumetric moisture. So, this has to be measured in the volumetric fashion. Now, these are the two thoughts of a school. One talks about gravimetric, one talks about volumetric. Now, tell me, in today's world, in 21st century, when you are using these concepts, which one you like to use and why? This is the age of electronics, sensing, clear? I need not to take a sample from the ground and bring it to the lab. By the time I bring it to the lab from Rajasthan, everything is lost. It moisture gets evaporated, agreed? So, what I want to do? I want to use a sensor. I want to install it there itself. What sensor is going to do? It is going to measure volumes, clear? This is becoming obsolete because lot of problems. First of all, take undisturbed sample, bring it all the way to Bombay in your laboratory. Their temperature would be 45 degree, 50 degree. How would you maintain the moisture of the sample? Very big question, clear? So, nowadays, it is good that you are asking this question. I am sure you are getting exposed to a lot of interesting ideas. So, the whole thing is you use sensors, install them in the field, measure this. Most of the automated sprinkling systems which are working are based on measurement of volumetric moisture. One more thing for gravimetric and volumetric moisture containers. If you do the experiment, then you will come to know the temperature that is used for gravimetric moisture content and at that temperature most of the organic matter also get vanished. So, explain this again. So, what he is trying to tell you is that why gravimetric moisture content should not be used when you are dealing with the soils because you have recently studied that soils have lot of organic matter. And if I expose the soils to high temperatures, I may lose the organic content and hence the results will be wrong. And this is the reason why people are not adopting the gravimetric methods these days, alright. So, please make two corrections. Gamma W upon gamma D upon gamma W into moisture content which is gravimetric and VW upon V. Now, let us go into the gammas. What is gamma total? Suppose there is a soil mass like this. This is the ground level and I want to find out what is the total unit weight of the soil at point A. There are several techniques. One of these techniques is, you know, we call it as a core cutter method. I am sure there is a laboratory experiment on this. You must have done this. What normally we do is we make a small hole over here on the ground, alright. Scrape out this soil, remove this soil, put it in a polythene back so that the moisture does not get lost. This weight of the soil which you have removed from here and if I measure the volume of the cavity which I have created over here, alright. This can be done with the help of sand replacement. So, core cutter method and sand replacement methods are normally used to obtain gamma D value. The procedure is like this. You take a small, you know, pre punctured sheet of metal, keep it over here. There will be a orifice in this and through this orifice you drop sands, clean dry sands. I can measure the volume of the sands which is filled in this cavity. I know the specific gravity of the sands. I can compute the volume, clear. I will tell you how to do this. The soil which you have removed is taken to the lab, dry it in the oven, get the weight, obtain the moisture content and that is what is going to help you. Normally core cutter method is done in cohesive soils. You will be surprised to know that despite all this technological advancement when it comes to the payment of the contractors, the only method which is used to establish the compaction, adequacy of the compaction is by core cutter method. There is no other way. Nowadays, what people are doing is they are also using different type of nuclear density gauges. Nuclear density gauge can be utilized to obtain the moisture content and the gamma D directly. There is another method which is known as a balloon method. This is the history of geotechnical engineering. People used to put a balloon inside and they used to pump air. The balloon used to take the size of the cavity which you have created and I can compute what is the volume of the cavity. I have taken out the soil. I know the weight of the soil which is displaced from here. I know the volume gamma D is known. Those of you who are interested in reading what is the state of the art on moisture content measurement, please go through the papers written by Socialakshmi and myself about 2 years back in the journal measurement. So, all the recent techniques of moisture content measurement of the soils are listed there. Whenever you get time, please read that and that talks about the historic development and what is being practiced right now. So, I am now coming to your question slowly, all right. So, whatever sample you have taken out from here you weigh it, you know the volume, you get the gamma D. Now, gamma D and gamma dry, they are related to each other. So, if I normalize gamma D with 1 plus moisture content, this is equal to gamma dry, clear. You see in the soil is such a tricky material that everything is possible on paper, but in field nothing of this sort is possible. So, I am sure the moment you scrape it out, it is not a intact lump, you will be very lucky if you can take out a core intact sample, it is very difficult. So, what you are saying is also possible if you are extremely good at experimentalist and the researcher and you value everything up to 0.3 decimal place, everything can be done, no doubt about it, but very difficult to do in the field. So, what we do is we measure the volume and weigh the soil, clear. You can do other way also, that is not a problem, but that is difficult. So, this is the relationship between gamma D and gamma dry. So, I think we have sorted out now gamma total, we have sorted out gamma dry and submerged is nothing but gamma D – gamma W. Now, one thing which is still pending is saturated. Now, so how would you go ahead with this? Can you solve these expressions? Try to derive now, normally I cannot remember these things I have to derive. So, this will be gamma W G plus S into E over 1 plus E, try to prove this, is this correct Vinny? Sure. So, using these functions, try to come to this, this is your homework. I will read out the expressions, gamma total is gamma W unit weight of water multiplied by G is the specific gravity of the soil, if you remember, learn the method of finding out specific gravity you might be doing in the laboratory, S into E, S is saturation multiplied by void ratio over 1 plus E. 1 plus E is known as what, sorry, you are right, this is what is known as specific volume, specific volume of the soil, clear? One of the ways of philosophically looking at these expressions is that you are normalizing the weights with the volume and I will show you how it is done. Is this part clear? 1 plus E is the specific volume of the material. So, truly speaking, this whole term is a weight term divided by the volume. Now, I can create one more condition out of it. Suppose if the soil is dry, see you are talking about this situation, so when you are dealing with a dry soil mass, S is equal to 0, what happens to gamma T mathematically? G upon 1 plus E, I will remove the brackets now. So, truly speaking, this is G into gamma W over 1 plus E, fine? One of the conditions, when S is equal to 1, this is saturated soils, okay, what is going to happen now? So, your gamma saturated, this will become gamma dry, is this okay? Gamma T has been replaced with gamma dry, so this becomes gamma W G plus E over 1 plus E. I hope you will not have much problem in deriving this, what you have to do is maybe as a rule 30th, when I am not here, you treat this as a practice session and it should not take you more than 1 hour, get conversant with all these things is so easy, alright? So, this matrix is like this, now you can derive this, we have talked about gamma dry, one more way of gamma dry was this, defining gamma dry was this, correct? This is normally done when you are dealing with the experimental work, either laboratory or field, so the best thing is you take out the sample, measure the gamma T, we never say we are dividing with 1 plus W, I call this as a normalization, this is part clear? I am normalizing something which with something and this happens to be your unit specific volume, 1 plus W is also sort of a specific volume in gravimetric form. So, this is how we normally treat gamma D, are you happy? This was your question, how to compute gamma D, gamma saturated, gamma submerged, gamma total, is this okay? Any other question? This is a state where AC is going to be 100%, what is AC? Air content, this is a state where AC is equal to 0, clear? Now, what you should do is, start from the first principle, keep on substituting these values and see what are the values you are going to get. Your initial hunch was that you are talking about the porosity if you remember, so now I am giving you a lot of homework, what you should be doing is use this function and see whether your hunch was correct or not. But as we discussed earlier, your air component whites will have water also but I am now creating a situation where there are no whites, clear? Your question of hygroscopic moisture content is a very tricky question, when I am saying gamma dry, this is a bone fried soil, a gross injustice with the soil which should not be done but it is being done. So, this is a shift between the conventional and unconventional new, new geotechnical engine. So, all this G which I am using, if you remember in the previous lecture what I said is, normally G is defined as G skeleton or G of minerals, clear? But for the sake of simplicity, we do not write G as we already there are so many times we are using S saturation and all those things, just for the sake of simplicity. Good question. So, G is meant for the minerals which are present in the soil solid phase, skeleton, clear? If a soil is multi mineral soil, then you have to go for a composite of G weighted average of G value, mineral phase number 1, specific gravity multiplied plus G1, G2, G3, G4, this divided by total weight God knows, clear? Fundamental clear? Now, what you should do is try to prove, try to prove this. So, I will tell you the philosophy part, this is a relationship which is known as phase relationship. Now, if you look at the left hand part of the equation, can you recognize what it is? And what this part of the equation is, you get the answer. I think we were discussing if you remember this versus this, volumetric versus this, what is G? It is a volumetric phenomena or a gravimetric phenomena? What is G? How do you obtain G? It is a gravity, read this, gravimetric phenomena, W again gravimetric phenomena, clear? Weights, volumetric phenomena, volumetric phenomena, biggest flaw in geomechanics, but this is the starting point. What you are doing here, unknowingly, you are matching gravimetric with correct, big question mark. If you answer like this in an interview board, they will be very happy that oh, she is the most great of the artist. And then comes all this RND which is being done, you know, whatever was done in 1930s, 40s and what is happening right now, there is a big shift. But forget about this question mark for the timing, as far as your 30 years old becomes concerned, this is the Brahma Vakya, we have to follow it, try to prove it. So, G is normally obtained by picnometer bottles, you must be doing it in the lab, go to the website, read how G is obtained, picnometer bottles, weight displacement. So, they take a bottle like this, these picnometer bottles are like this, you weigh this weight of the soil mass, put it inside, weigh the whole thing, fill it with water, the equal volume of the water has been displaced, all right. You measure that, that is one way. Second way would be weight of a material in air and divided by weight of the material in water, clear? Two methods. Third method is more complicated, but what people like researchers use, so this could be gravimetric also, displacement of water, these are known as gas picnometers. We use these type of equipment in our laboratory, where the helium gas is replaced by the material and we measure the volume of the gas replaced. Why? Because helium is a conservative gas, it would not react with the material and hence the chemical activity of the material remains intact. Another thing could be, there are several other methods like thermogravimetric analysis, TGA, which chemical engineers do, they find out the specific gravity of raisins and so on. So, there are several methods, but for the time being, please read this, this and forget about this, okay. So, what is going to happen for saturated soils? s is equal to 1, that means g into w equal to e, it is an interesting relationship, but for a very specific case when the soils are saturated, all right. Gamma submerged you try to derive as gamma total minus gamma water and this you should be getting as gamma w g minus 1 over 1 plus e. This expression we will use later on when we are discussing the seepage theory, all right. I am sure you can realize that gamma submerged divided by gamma w, all right, this becomes a non-dimensional term. So, this is the normalized submerged unit weight of the material which is g minus 1 over 1 plus e, interesting philosophy behind this, g is the skeleton of the material of the soil, minerals, clear and e is the void ratio. So, this is the term which can be utilized in defining the stability of the structures. The effect of unit weights gets filtered out. So, one of the ways of looking at these expressions is how are you normalizing something with what, this is okay. How this expression is used for determining stability of structures? Write down this question, we will discuss this when we start discussing on the seepage theory, all right, stability of the dams and all. So, then we will use this as a critical hydraulic gradient, correct. So, if that is the situation where I am trying to simulate the dancing sands, liquefaction, you know the effect of the weight has gone, gravity does not affect this material now. What has happened? The material in itself is now behaving the way it has to behave in a matrix. This subject is more of a philosophy rather than mathematics, is this okay? So, a specific gravity of the minerals, the heavier the minerals, this system is going to be more stable, organic clays, what is going to happen? God knows, G is extremely low, we will talk about this. Void ratios could be very high, marine clays, I said very sensitive material, very high value of this, very low value of this, very susceptible to decay, forget about hydraulic gradients, decay, they just get decayed, all right, is this okay? So, the way to look at these expressions is try to put a lot of physics behind this, okay. So, I think this we have already done gamma t and gamma dry. So, another interesting thing is you might, you should learn how to change gamma t to gamma dry also. So, the moment I put s equal to 0, this system becomes, you know, one of the ways to convert this to gamma dry would be, I will use now gamma d henceforth, is this okay? So, gamma d would be substitute s equal to 0. So, this becomes G into gamma w over 1 plus e. Until now, what I did is, you know, I was dealing with this three phase model like the way I was dealing with it. Now, suppose if I say I define v as 1, you think concept will change? Until now what we did, we were defining v as vv plus vw, clear? This is another perception of looking at the things, that total volume I am assuming to be unity. So, that I can get rid of 1 plus e thing, which I use as a specific volume. It is a cumbersome process, you know, getting e value, w value, there is so much of discussion which we did of how to get the water content, how to get the void ratio, how to get the porosity, this, that, what not. Get rid of all this. You assume that the total volume of the soil is unity and then back compute everything. How will you do this? Any idea? So, what will happen to the porosity now? See, your this system is now 1, all right. This is another way of looking at the things, yes? Oh, sorry, sorry, sorry, yeah, you are right. This is vs plus vv equal to 1, correct. So, the total volume is 1, yes, thank you very much. Keep correcting. Now, what is going to happen? The whole story will change. How? Any guess? I am talking in terms of now volumetric things. Earlier we were discussing everything in gravimetric things. Write down these final expressions and try to prove them. You see, your porosity would be vv upon v, this will become vv and hence vs will become 1 minus porosity. So, your ws will become g into gamma w 1 minus porosity, where w is w weight of water upon weight of solids. So, if you substitute this whole thing, what you will be getting is this will be equal to eta into sorry s into e into 1 minus eta. Try to prove this function and of course, gamma w will come. Here, what has happened is this three phase system if you remember with which we were working v air, v water, v solids, this whole thing has been treated as unity. So, I am doing now calculations which are volumetric, volume centric, you may say. Try to prove that gamma is equal to g into gamma w 1 minus eta into 1 plus w. So, gamma drive will be equal to 1 minus eta into g into gamma w. And what about the saturated thing g gamma w 1 minus eta plus eta into gamma w. First one is, yeah, this is gamma total. This will be interesting for you to derive. I expect all of you to derive these expressions. Another thing is derivation does not mean that you manipulate with this term and show it equal to gamma t, fine. It also does not mean that you substitute the values of some parameters and say that this is equal to gamma t. It should be always starting from first principle. In short, what has happened? As I said, there are two ways of looking at the things. Look at this function. You know, we very conveniently got rid of the volume of the pores, which is very difficult to measure. So, when you get a chance to do high studies, then you will realize that the volume of the pores is measured by using lot of sophisticated instruments like SEM, MIP. This is beyond your purview. The scanning electron microscopy can be utilized to compute the volume of the pores. Remember the pictures which I showed you in the last lecture and I showed you that there are cavities between the minerals. So, those cavities can be scanned by using SEM, though it is very difficult or you have to use mercury intrusion porosimetry. This equipment is available in our lab. Those of you who are interested are most welcome. The entire country uses this equipment in our lab. You intrude mercury at a certain pressure and get the pore sizes. In short, it is a difficult task to measure volume of the whole soil mass. But what you were saying is correct. You can take out the soil and put it in a mercury pot. And this is what is known as displacement. So, you find out what is the volume of the mercury which got displaced, that is the volume of the solids and then you can go ahead with this. But again a difficult process, clear? What we have done is mathematically we have got rid of the volume by assuming this as unity. This is the beauty of this system and then we are saying that the volume of voids is same as the porosity, clear? And from here we can compute the volume of solids as 1 minus porosity. This has its own advantages but delicate thing to handle. Just for the academic interest, you should be aware of what are the specific gravities of different type of soils. You may say Gs, what I have been using all the time as G quartz is 2.65, illite 2.84 basically it varies from 2.6 to 2.65, 2.86 sorry, then montmorillonite 2.74. These are peculiar numbers and these numbers indicate prima facie what is the soil with which you are dealing. So, in medical profession this is your diagnostics, clear? You need not to do much, take the soil put it beneath XRD and you can quickly get the mineralogical composition and you can establish what type of soil this would be predominantly because later on now you will see what is the art of consulting. First to understand the material, whether these mechanisms are going to take place in this type of material or not. So, I get lot of information from these numbers, highly stable system for your information and highly unstable system, clear? I have the strategy in my mind how I have to treat this patient, highly stable, highly unstable. The poor surface area, a very high surface area, very high cation exchange capacity, very high water absorption capacity, clear? Extremely low everything, very poor cation exchange capacity, very poor water absorption, very poor surface area. So, the moment you look at the first number in your report of the material you have a strategy and you can catch the person, by the way this profession is becoming more of forensic in nature. Forensic means first thing you have to establish is what is the material which was utilized for creating something and you can prove that this is a fundamental mistake, agree? So, looking at these numbers I can make out lot of things about it. Just to give you a feel of what peats would be, if you remember peats are organic substances specific gravity is 1.26 to 1.8, humus 1.37, peculiar numbers, bentonite 2.34, clays mostly would exhibit 2.68 to 2.8 and just to give you a feel of silica fume which is man-made specific gravity would be something of 0.5 to 0.7, very good admixture for concretes it gives you lot of durability, is this okay? The best way to analyze this would be take help of XRD which I was discussing the other day in the class except diffraction, you remember or sometimes those who are very good in chemistry what they will do? They will do stoichiometry to find out what mineral is present. So, if you go to the organizations where they are dealing with the production of coal, they keep a strict vigil on the type of coal which is being produced, you know why? It gets reflected over here and the cost of the coal depends directly on inversely proportional on this specific gravity. Are you realizing this? What is the steel industry? So, when you make sponge iron, when you make different type of iron, cast iron, specific gravity is very, very important. What is the specific gravity of steel? Where it would set quick, sorry, very good 7.8. So, steel is somewhere 7.8 at the extreme end, alright? So, the blast furnace slag which you are using for making different type of concrete and all and suppose if I mix them with the soil what is going to happen? The system is going to become denser and hence I am stabilizing it, clear? Concrete is nothing but combination of these minerals. So, the moment you add GGBFS slags, it becomes denser, it becomes more durable, more strengthful and so on. Is this okay? Do not mug up all these numbers and all, okay? But just keep it in your mind, that numbers tell you a lot. Let us spend some time on now the grain structure again. Having done all this, particularly the granular soils, if you remember in the PowerPoint presentation I showed you that most of the granular structures are single grain structures and we normally depict them as perfect spherical entity. But I am sure if you look at below my microscope even, you will find that they are very distorted. So, there is nothing known as a standard sand which normally is used in the practice that this is a standard sand. Standard sand is also going to be quite irregular in shape but we assume this because this is how these sands are produced. So, normally standard sands are used for finding out the in-situ gamma t, agreed? You remember that corkutter method or soil displacement method you might call it sand displacement. So, you use standard sand. So, granular soils are normally depicted as perfect sphere and they have two types of structures. One is what is known as cubic. So, a cubic structure would look like this, the front view or the plan, it does not matter. Can you compute the wide ratio of this system? Compute E and compute porosity, fine? Keep doing this, this is all homework for you. The second thing is now this is what is going to be the looser state of the material where E is corresponding to E max and eta is corresponding to what will be eta now? Porosity, max or minimum? Max, very good, say it loudly, why? I am happy that you did not get confused though I tried to confuse you. Now, this is where the concepts are important, you are getting this point. This material is highly permeable, it will allow everything to pass through it, very good filters and so on whatever we discussed there. At the same time, now what is going to happen, earthquake comes, is this a stable state of the structure or the structure of the soil or the grain structure of the soil? No, I hope you can understand. So, in case of any lateral motion, lateral stresses, shearing of any type, what happens is this gets converted to, what will happen? Suppose if I depict the plan view like this, please excuse me for my poor art and on the top of this will be the second series of the grains, it will be sitting like this in the void space, can you imagine this? So, what is going to happen in the side view if you see, what type of structure is this? Ceramic, pyramid, yes, you are right, very good. So, this is a pyramidal, sometimes people also call it as a rhombic structure, pyramidal, correct. Now, what is the mechanism of transformation from this state to this state, have you understood? So, this is going to be a pyramidal structure, people who are coming from the Gangetic belt parts of Northern India, you know, close to Bihar, UP and all these places, single grain structure, agreed, gravity is predominant, these particles are coming in sediments all the way from Himalayas, because of the velocity of water, they got grinded enough, water is the most abrasive material, what it did, all sorts of irregularities taken care of, the particles have become spherical, shining particles remember, they would settle down slowly and slowly, how they will settle in water body, first layer, second layer, third layer, loses possible form, it is okay, have you understood this? Dangerous situation, you cannot construct any infrastructure on this. The moment earthquake comes, what is going to happen? Because this is not a stable state of the material, you can use entropy of the particles to make sure that this is not a stable system because the entropy is going to be more or less here, read this. So, those of you who are going to master this concept would go into the particular mechanics by using the concepts of entropy associated with the particle system, particular mechanics is going to be more, any system cannot remain at higher entropy, small shaking up is going to cause rearrangement of the particles in this form, which is more stable. So, you are very happy with this restructuring or not, life is not so simple. So, even if you densify the materials from this state to this state, now this is a practice of geotechnical engineering remember and again earthquake comes on this system also, what is going to happen? Poor pressure is okay, that is fine, poor pressure will never develop in coarse-grained material, remember, please remove this from your mind, coarse-grained material, poor water pressures will not come into the picture, clear? This earthquake loading is for a few seconds and within that few seconds the permeability of the system is so high that poor water pressures are not going to build up over there. Now, what is going to happen is the moment this type of situation occurs, look at this, tomorrow you go to the hostel number, I do not know in which hostel you have a snooker table, try to stack the snooker balls like this and press them like this, look at the hand motion, what tectonic motion does earthquake comes, everything oozes out, ice cream scoops are a beautiful example of learning how the state of stress develops in the system, alright? All type of haulers which you use in civil engineering construction, they go this and they push the soil up, ice cream scoop is just like this, is it not? You push it and then figure out, snatch it, so that means if again earthquake comes, what is going to happen? The rolling starts, but the intensity of volumetric deformation from this to this are less and from this to this are less, lesser, so the whole concept of modifying the grounds for making infrastructure would be to create a state of the grains, particularly in the granular size which is more stable. Now, this concept we will study in the second course, have you followed this? Now, this is going to give you, please compute now the porosity and this is going to be E minimum, minimum possible state of the void ratios and the porosities are also going to be minimum, clear? I will give you quick numbers so that, but try to prove them analytically. This is the cubic structure and this is the pyramidal structure, where the balls are coming and sitting over the hollow spaces, this is how it looks like in the side view, is it okay? This is the first layer of the particles, first layer on the top of this, the second layer comes and gets deposited, second layer and third and fourth and so on, it is okay? So, typically for sphericals like this, the E max value is 0.91 and porosity is 0.34, E minimum is 0.476, these are the magic numbers 0.253, try to prove this, very unstable state, so called stable state, but still it is under threat because of the dynamic loading, pore water pressures will never develop, is a granular material clear? So, understand the material first, but yes, if you have fines into it, somehow impregnated, yes, you may think of, we will discuss about that slowly. Now, there is a concept of, somebody may ask you, is it possible that void ratios can be greater than 1? Is it possible to have more voids than the solids? Are you sure? Sure, negative marking, your answer is correct, it is possible, but only nature can do this, remember, we cannot do this, we cannot reconstitute the soils with void ratios more than 1. Now, nature is a great engineer, what it does slowly is, it produces the deposits where the void ratios could be more than 100%, extremely soft sensitive deposits, Aksa beach, beautiful example, they put a warning during rains, nobody should go there, why? The chances of drowning are more, not in the water. So, imagine this is a state of material where the voids are more than the solids, you are right, absolutely right, clear? We define a concept relative density of the matrix of the soil, remember, we call it as Rd. Now, in mathematical terms, Rd is defined as E max – E divided by E max – E min and of course, multiplied by 100, apart from the mathematical expression, the way to understand this would be truly speaking, you are trying to find out the deviation of the void ratios in 2 states of the material, clear? One is very loose state and one is very dense state. So, this becomes a benchmark of that of the torture which a material can be subjected to. This is a torture, clear? You have disturbed the whole thing by torturing it. So, E max and E min, the difference between the 2 becomes a scale which is used to normalize the state of the material, this E is in C2, another new term which I have used today. In C2 is as it exists in the nature, clear? XC2, once you have taken out the sample, you have disturbed the as in nature state, we call that as XC2. Bring this sample to the lab, you can compute E, gamma T is known, dry it, gamma D is known and gamma D is G into E over 1 plus E, not G into E, this is G upon 1 plus E into gamma W, clear? So, from there, if you know the void, if you want to know the void ratio, you know the gamma D, you know the value of G, do test, get the value of E, substitute it over here. So, this becomes infield. The best way to get E max E min is take a small cylindrical ring, pour the samples very gently, volume is known, weight of the material can be obtained and from there you compute E maximum, shake it on a shaker table, let it settle down, compute the volume change which has occurred, measure that volume, weight is constant, you get the E min value, this is how it is done, fine? Now, this RD is an indicative of all the engineering properties of the soils, there is a classification system based on this. Like as you said about the E max, how we find the E min? No, what you have to do is for E max and E min, there is a, we call it as a standard ring or a cylinder. So, you take a cylindrical unit and in this unit, you just pour the dry sand, you know the initial weight of cylinder, you pour the sand up to the brim, stop the process, you know the weight of the cylinder plus weight of the soil, you know the volume of the cylinder, internal, you get the gamma D value, G can be obtained, E is known, so you get E value, this is how you are getting the maximum value, E max, shake this whole thing on a shaker table, you get E min value, all right? Three things, so the field sample which you are going to bring is E. Now, try to prove this relationship RD equal to, now can you derive RD as a function of let us say gamma max, gamma min and porosity max and porosity min, based on the RD of the material 0 to 15, 15 to 35, 35 to 65, you should try to remember this though I cannot, I am sorry for that, but this is going to be useful for you, so this is very loose state, this is loose state, this is medium dense sand and this is dense sand, so for the practice of geotechnical engineering, these numbers are required, do not bother much, all this is available in the course, so when you practice a subject, you have the bibles with you, is it not? And the bibles are nothing but the quotes, this is the container, I will keep on pouring the soil very gently from a lower height, let it get filled up, I will stop the process, I will weigh the soil, I know the volume, gamma is known, it is a dry material gamma D, G upon 1 plus E, G is known, E comes from there and that E is corresponding to E maximum, because they are loose state, here, so if you do like this, what happens, it gets densified, I can put on a shaker table, that is a better scientific way of doing this, I can fix the amplitude, I can fix the number of frequencies whatever, so this is going to drop down in volume, so I know the new volume, I know the weight remains constant, I know the specific gravity, now this is what is going to be your E minimum, clear? So this is known and this is, you take out the sample from the field and good.