Hello everyone. Let us summarize whatever we have learned so far. In the fundamental principles, we have learned that there are constituted relationships, state variables, and material constants, for any given phenomena to define. To define any given phenomenon, we need to understand and distinguish what are the state variables,constitute relationships material constants. The governing equations in continuum physics are basic principles that represent fundamental physical laws that are independent of matter or independent of material. So, they are the same for any given material any given phenomenon such as conservation of mass, conservation of momentum conservation of energy etcetera. The dependentvariables in these governing equations are the state variables. Because they represent the state of the system; however, we require additional equations to solve a given problem or to mathematically solve a given phenomenon. Such additional equations that areconstitution dependent are called constituted relationships are depending on the material. Material dependent equations are called constitutive equations. Constitutive equations, therefore, are a relationship between 2 different state variables; such as a relationship between void ratio and effective stress, or a relationship between effective stress and shear stress. So, the equation constants in these state variables are called material constants. So, we required to understand the middle constant dependency on other state variables also. So, once we identify what are the different constitutiverelationships for a given phenomenon and establish the relationships and identify the middle constants, we can understand the phenomenon. So, this is the flowchart for the phenomenon prediction is like this. So, this is a physical phenomenon or observation. So, from this after observing them physical phenomenon we frame governing equations. For example, we can think of a consolidation settlement. This is slurry when a tailing point the mine tailings would settle with time. So, the phenomenon here is the settlement of mine tailings with time. So, is the consolidation settlement? So, to predict that phenomenon we need to developgoverning equations. So, such as we could utilize the basic physical laws like conservation of mass, conservation of momentum, or force equilibrium, and energy conservation. So, here a consolidation settlement. So, from these governing equations, we identify thestate variables. Maybe the density or any other state variables we identify. So, from that, in this particular case the state variables are maybe the void ratio is reducing. Because the volume is reducing so, the volume is void ratio is decreasing with time. So, therefore,void ratio is one state variable. The stress state would differ with depth may be the effective stress is varying with depth. So, you have another state variable like sigma hash. And pour water pressure also changes with time. So, therefore, it could be another state variable. So, once you identify we can develop constitute relationships that provide interdependency of different state variables. So, this could be a relation between void ratio and effective stress. So, this is the void ratio versus the sigma dash. Or it could bethe hydraulic conductivity also changes with depth. So, hydraulic conductivity is amaterial constant, but it could change with the void ratio. So, you may require another relationship such as this. So, once you will have the constitutive relationships, we have we identify the materialconstants. The material constants could be a v or m v or could be your c c etcetera. So, once we have this again or could be c v also. So, these are the material constants. Once we identify these constants, now we can predict the phenomenon. The phenomenon issettlement versus time. If the phenomenon does not, then we compare this phenomenon with the physical phenomenon. If it does not match, there will be more constitution relationships required.Therefore, so, by providing those constitutive relationships we try to match the prediction try to get a good prediction of the phenomenon. So, for example, the onedimensional consolidation can be Terzaghi, which uses only conservation of mass, and which uses Darcy’s law also; which combines these 2, and it provides a governing equation, which creates the governing equation the state variables that it considers is pore water pressure, sigma dash, and e dash. So, the constitutive relationships are e dash versus sigma dash, which are assumed to be constant at any given change in the effective stress. And it uses k versus e relationship. The k is assumed to be constant, for a given increment of the loading, k is assumed to be constant. And which drives the coefficient of consolidation. And, from the phenomenon, we predict the c v value. And we match the governing equations are we ma match the development model with the physical phenomena observation to determine the c v. This coefficient of consolidation is used in the field for estimation of consolidation settlements the rate of consolidation settlements. Similarly, in unsaturated soils also we need to identify are different state variables so that we can develop governing equations and it can develop constitute relationships for understanding the phenomena. In unsaturated soil mechanics, we have seen that we have 3 phase system; where we have water air water and solids. So, this is solids, and you have air and water. So, you have volume ratios generally consistent on this side. So, volume fractions are considered generally in this side, volume of air, the volume of water, and volume of solids. And this is the mass of air which is 0. And the mass of water and mass of solids. So, the air phase generally defined as the poor space which is not occupied by water. So, there is called air phase, and the water phase is the poor space which is not occupied by air; however, the air phase contains the air phase also contains water vapor, air phase contains water vapors. And the water phase contains dissolved air. As a solute, you have dissolved air. And the phases are no means pure; the water can exist in the air phase as water vapor. And air can exist in water as dissolved air depending on different conditions, like pressure-temperature etcetera. So, solids are generally the range of solids varies from very fine-grained soils such as clays silts and organic matter to coarse-grained soils like sand and graphite. So, the air phase when you consider, the air phase has certain state variables, such as the density of air, which is an important state variable that governs to flow through unsaturated soils. So, the density of air is an instrumented from this particular equation. Here u a is the airpressure, the molar mass of air, R is gas constant T is temperature. So, the molar mass of a substance is a mass of a substance divided by the amount of substance, which isexpressed in kg per kilo mole or gram per mole. And u a is the mass at a pressure which is expressed in kilopascal. R is a gas constantwhich is 8.314 joule per Kelvin mole. The temperature is in Kelvin. And the density of air we have estimated at a standard temperature of 25 degrees, and 101.325-kilopascal pressure. This value is 1.185 kg per meter cube. The density of air is sensitive to the pressure and temperatures. So, the variation of density can be represented by themathematical equation like this. So, the density of air increases with an increase in air pressure but decreases with an increase in the temperature. As with the increase in the elevation from the mean sea level the pressure decreases. The atmospheric pressure decreases therefore, the density of air decreases. This was explained by simple illustration where if you fill the water bottle with air at a higherelevation, close it with airtight and bring it to lower elevations or bring it down from the hillside. And when you bring it down because the air pressure increases in the surroundings, this can crush the water bottle, this was explained during this discussion, this was explained earlier. So, the temperature increases the density decreases, this is this could be understood from sea breezes we see on the shores. Because the land quickly gets heated up, but the air above the sea relatively is cooler compared to the air above the land. So, therefore, the air above the land density decreases, it gets uplifted, this is filled the space is filled by the air which is coming from the sea. So, you get a cool breeze. So, this is how the air circulates. Coming to the water, this is air, coming to the water. If you look at the water density, water density has a direct influence on the physical and mechanical behavior of unsaturated soils. The waterdensity varies with temperature in this manner. As the temperature decreases the volume of water decreases. So, therefore, the densityincreases, and the richest one critical value and beyond that the volume increases our density decreases. So, this maximum density which is achieved at the lowest volume which is achieved is at 4 degree Celsius. And which is, 1000 kg per meter cube. Theseare on 9 9 6. Sorry, an increase in the volume below 4 degrees is called the anomalous expansion of water. Because of this, the ice floats on water. The ice which is lesser denseis less conductive to temperatures; because of this, the aquatic life survives in the coldcountries. Because once the ice forms, and it floats at the surface ice is less dense compared to waters colder than this compared to waters. Because of this phenomenon anomalous expansion, and when it floats it will act as thermal insulation are partially thermal insulation and it does not allow heat to transfer. So, that beneath the ice the temperatures are maintained at higher values. Because of this, the aquatic life survives. If the whole thing freezes continuously then aquatic life cannot survive. Apart from the dependency of water density and temperature, the water density significantly changes due to physical-chemical effects. We have seen that the clay particles collide, they are plate-like particles which has a negative charge on the surface due to isomorphous substitution. Therefore, there are positive ions acquired from the environment during the formation orthe chemical weathering process. And they are present at the surface. So now, when the water is available the water hydrates all these cations, and also hydrates internal lattice and also external surface. When this water absorbs, the water is available is a thin filmaround individually clay platelet. This water content can be as high as 15 percentage at edge rate state for one molar at rich clays, this was explained earlier. So, therefore, water as it got absorbed on the surface of the clay, which is tightly held on the surface in an absorbed form. The density of water is different from the free state.So, if we can draw the density of water or water density in gram per centimeter cube, and the gram metric water content on the x-axis. It is observed that water density can increase to very high values. This value 1.4 and this is 1. So, as the water content decreases when we are going in this direction water content decreases the density ofwater increases; because the water which was held in the absorbed state has very high densities. The densities can even reach up to 1.8 gram per centimeter cube. They form on more right rich clays, this was recent observations given by Luke in 2018.
Other than the density of water and air there is a viscosity of air and water. The viscosity of the fluid is defined as the ability of fluid to deform under shear stresses. So, the hydraulic conductivity of water and air depends on or gets influenced by the viscosity of air and water respectively. The viscosity influences the hydraulic conductivity of both the air and water. So, therefore, the determination of viscosity of air and water is important. Dynamic viscosity which is represented with mu has units of Newton second per meter square or centipoise this is express the resistance to fluid flow. Another one is kinematic viscosity, which is exposed to the nu, which is the ratio of dynamic viscosity to the density, which is used in analyzing the Reynolds number. So, if we see the dynamic viscosity of water and air, and which viscosity for the water the dynamic viscosity decreases, but air viscosity increases. But for air the viscosity increases. So, this is for water and this is for air. So, this value at one particular temperature, it is values around 10 to the power minus 3 in Newton second per meter square. And at the same temperature, this value may be very small this is how the ranges are. So, mu of water at 20 degrees is 1.002 centipoise or 0.1 Newton second per meter square, and mu of air at the same temperature is 0.0018 Newton second per meter square. Air viscosity increases with temperature. As the temperature increases, the air molecules get additional energy and they collide with each other, therefore, the viscosity increases.On the other hand, the water viscosity decreases with an increase in the temperature, because the increase in the temperature makes the water molecules to overcome the intermolecular forces. So, this decreases the dynamic viscosity of water, understood the thermodynamic equilibrium between different phases. We have understood from a given beaker or a taken one glass of water. For example, if we take a glass of water. So, this is water. In this you have, water molecules, these water molecules which are available at the surface would leave the surface and go to the air phase because the intermolecular force at the surface is not strong enough to hold them back into the liquid. So, they go into the air phase. For example, if this is closed by a lead, the water molecules which going to the vapor are in the air phase would eventually have to come back by hitting this plate and again go into the water phase. After equilibrium, the number of molecules which are leaving from the water phase would be equal to the number of molecules which are coming into the water phase. So, at this equilibrium, for example, this is coming out and this is going up, this is leaving the surface. So now, the molecules which are actually coming and comingto the water phase are called condensation. And molecules that are leaving the surface are called vaporizing. So, therefore, the number of molecules which are hitting the surroundings or surface, thisplate the pressure which is exerted by these molecules, these vapor molecules on the lead surface is called vapor pressure. Are the number of molecules that are available in the space also can be defined based on the density? So, the number of vapor molecules is the mass of these molecules per unit volume can be defined as vapor density. So, this vapor density is nothing but the total humidity. And the relative humidity is defined based on the vapor pressures. So, based on the partial pressure of all gas components which are proportional to the molar fraction of each component; this is what we have defined, u i by u a is equal to n i by sigma n i. So, the partial pressure of each component is related to the molar fractionof those components. Or it could be expressed in terms of volume fraction also because the ideal gas volume is known that is 32.414 liter per mole. So, this vapor pressure increases with an increase in temperature because the temperature is increased. Moreenergy is given to the molecules of water, and they can overcome the intermolecular forces. And then they try to escape from the surface. Therefore, more water molecules exist in the gas phase, gas phase therefore, more pressure is exerted. So, therefore, the vapor pressure or vapor density increases with increasing the temperature. As we defined the total humidity is total absolute or absolute humidity sorry, this is absolute. So, this is absolute. So, the absolute humidity is the mass of water vapor in a given unit volume is called absolute humidity. On the other hand, the relative humidity R H is defined with respect to the partial pressure of vapor. So, this is the ratio of absolute humidity in equilibrium with some solution to the absolute humidity in equilibrium with pure water; which is expressed in percentage. So, if you take pure water, and the density of vapor which is present on the above this water pure water which defines the absolute humidity, and if you take saltwater or any othersolution, the humidity or relative density above this salt solution. And the ratio of these 2 the absolute humidity above the salt solution in equilibrium and the ratio of absolute humidity of the vapors above the salt solution and the vapor density above the purewater, and this ratio is called relative humidity.So, this relative humidity decreases as the salt concentration increases. And it alsodepends on the type of salt which is present in a given solution. And also it depends on the temperature. The saturated vapor pressure is defined to understand what are the general ranges of relative humidity because even though you have pure water nearby therelative humidity cannot be 100 percent. The relative humidity decreases due to the presence of some solute or presence of some salts in the solution are due to these factors the relative humidity decreases and to understand the variation the relative humidity.And the formation of dew points etcetera, we can require the variation in saturated vapor pressures with temperature. So, this is vapor pressure and this is temperature. So, this is how saturated vapor pressure varies. This increases exponentially nearly. So, thesaturated vapor pressure, this can be expressed as an equation in terms of kilopascal. This is called Teten's equation 0.611 times exponential of 17.27 T minus 273.2 divided by T minus 36. So, this is Teten’s equation and we have other equations also. We have I have discussed other questions also for fitting this saturated vapor pressure. Generally, the soil or in the atmosphere, the vapor pressure maybe somewhere here. The temperature of the atmosphere is this much, and the vapor pressure is represented by point A here. Point A here, now we can reach the saturated vapor pressure in 2 ways: oneby decreasing the temperature. So, we follow this curve, and we hit this one we get the saturated vapor pressure. Or if you increase the vapor pressure by keeping the temperature the same and we also reach the saturated vapor pressure line. So, more often we see that in the atmosphere suddenly when the temperature drops, from here to here, that is the formation of dew or condensation that takes place. Let us solve one simple problem to understand the dew formation. Which is, you have a soil sample. So, in the atmosphere the currently the relative humidity is 80 percent. When the humidity is 80 percent and the temperature is 22 degree Celsius. And thequestion is what is dew point temperature when the vapor density remains constant for the dew point to form are to hit this saturated vapor pressure line what is the temperature required. So, this information is sufficient to estimate the dew point temperature. First,we can estimate what is the saturated vapor pressure at this particular temperature. This temperature is 22 degrees Celsius, and this is dew point temperature which is not known.So, we can estimate the saturated vapor pressure corresponding to point A. So, this is B, the saturated vapor pressure at A or 22 degrees is equal to 0.611 exponential of 17.27 into multiplied by 22 divided by 259.2, which is equal to 2.64 kPa. So, we estimated what is point C? At point C the vapor pressure is 2.64 kilopascal; however, the relative humidity is known this is 80 present. Therefore, we can calculate to u v at A. U v at A is 80 percent of 2.64, which is 2.11 kilopascal. So, therefore, we know the vapor pressure at point A. If this is known, when the temperature is decreased the vapor pressure remains the same, but this vapor pressure on the vapor pressure at the dew point. So, the saturated vapor pressure at point B and vaporpressure at A both are same, because it maintains the same vapor pressure, but the temperature only decreases. So, therefore, we can estimate we can say that the vapor pressure at B that is dew point that is saturated is equal to vapor pressure at A. So, we have the saturated vapor pressure here at the dew point. So, that is 2.11 which can be equated to 0.611 exponential of 17.27 times. So, this temperature is not known. So, this is a T plus 237.2 in terms of degree Celsius if you write. And when we solve this, we can get T equals to finally, 17.07 by say 0.928 is equal to 18.4 degrees. So, this is 22 and this is 18.4 degrees. If nearly 4 degrees drop in temperature takes place, then dew forms. We see that the small droplets drop. You can feel the water droplets. So, this is dew formation. And this concept is very, very important in future understanding dew point potentiometer principle, which is developed for estimation of the total section of the soil. Which are the important state variables for unsaturated soil mechanics, and in the dew point potentiometer? So, the sample is taken, but the sample's initial humidity is not known. Only the temperature is known then the sample is placed in the dew point potentiometer. Temperature is reduced gradually so that they can find out so that the dew point can be found out using some physics principle. Once the dew point is found or dew point temperature is found, the existing relative humidity can be found. This is an inverse analysis of this particular problem the same problem. So, these concepts are very, very important for understanding the principles of several operators used for the estimation of state variables of unsaturated soils. Thank you.
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