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Module 1: Constitutive Equations

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Vectors and Tensors

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Vectors and Tensors
Welcome back everybody to another lecture on introduction to soft matter. And last time, we have started with discussing what are known as constitutive equations. There are very important class of equations, but they differ from the conservation equations. And I had said before that the constitutive equations are approximate and they can often be just phenomenological models for understanding reality. And what better way to see an approximate relationship than the example of the perfect or the inviscid fluid or the perfect fluid that we were discussing it.(Refer Slide Time: 1:11)So, here we have also started introducing the tensor and the vector notation. So, I would like to clarify that the notations that we will use here what we will do is since we have started using tensors, we are going to let us say you have a variable u, if we do not, if we just write the variable u then this will imply that it is a scalar quantity. So, for example, pressure here that is we did not without any bar or without any other accompanying symbols. But if you put one small bar underneath, then what, it will imply a vector for us. And the same variable if we put the double bar, then this will indicate a tensor.Now, just to clarify scalars are actually also considered zeroth order tensor. And a vector is also considered a first order tensor and a tensor is obviously the generalized form of all this.But in our work, it is important to distinguish that these different quantities between thesedifferent quantities. So, we are going to adopt this notation for the rest of this course. And theone of the important quantities we saw was the Kronecker delta. So, what is, we will definethis also.Now, Kronecker delta is defined as a dot product of 2 vectors. So, I am going to put 1 barunderneath each of these and these are unit vectors in ith and the jth direction. Now the ithand jth direction independent of each other and hence, so this product is also called ij  andthese are all for Cartesian coordinates. So, we will take i comma j as varying from 1 taking onthe values 1, 2 or 3, we are talking about 3-dimensional space.(Refer Slide Time: 3:29)So, this quantity, so this quantity can take on different values and if i and j are the same, ifthey time stand for the same axis, then what will happen is this is a dot product of the same 2unit vectors in the same direction. So, it takes on a value of 1 when i is equal to j and when iand j are 2 different vectors or vectors in 2 different directions, then they are also independentof each other. So, the dot product is always 0 and i is not equal to j.So, we are getting started with notation. And another important point to be noted with this,what we will be doing is we will be adopting what is known as notation where a repeatedindex basically implies submission. So, for example, so we have written here ij  , i and j are 2different indices. But, we can also write let us say there arises a situation where A is a tensorand when written in component form let us say it is ij and then it is being multiplied withanother tensor, let us say B and this now has 2 components which are indicated as j and k.Now, you can notice the j is the repeated index here. So, this implies that this entire quantityhas to be summed over all the possible values of j.So, this is basically implying that this quantity is this shorthand expression is the same as thissummation. And now, since we have already said j goes from 1, 2, 3, we can write here, jgoes from 1, 2, 3. We will keep our attention focus on the 3-dimensional space, so this isgood enough for us. So, that brings us to let us, so we discussed the topic of the inviscid orthe perfect fluid and you can probably see the chain of thought we are adopting here. Whatwe are trying to do is we introduce the idea of constitutive relationships and we arediscussing different forms of constitutive relationships.But you probably know already where we are going to go, we this is a course on soft matters.So, we are going to look at different constitutive relationships that can hold for viscoelasticmaterials. And, we are to build up to that, we are trying to look at very-very simpleconstitutive equations and try to understand those in the process. So, by the end of this class, Ihopefully, you will be able to get to a point where we will discuss some of those importantconstitute relationships that you will require in your study of viscoelastic materials.(Refer Slide Time: 6:43)So, the next constitutive equation will be Fourier’s law. Now, if you have taken a heattransfer course, this is something that you probably would have seen quite a bit by this timeand Fourier’s law, if you are already familiar with it, you probably know that it relates whatis known as the heat flux with the temperature gradient. So, q = −k So, this  here, thisdenotes the temperature field and k here denotes thermal conductivity.I did not write a bar underneath it and the reason for that is it is a scalar quantity for isotropic materials, but if the material is not isotropic, then this k can also become a tensor itself. So, we just because we are just trying to discuss more one and more complicated equations, we do not want to really get into, our objective here is not to understand Fourier’s law, so we will leave it at this. The next one is what is known as the famous Hookean solid. Now, the concept of Hookean solid obviously comes from the name Hookean solid is because of being associated with Robert Hooke who wrote a very famous original paper, I believe the year was 1678 and this idea has been developed quite a bit after that.So, after Hooke’s original papers, this idea was developed by many, many, many scientists and around 1800s it was Cauchy who gave the form of the proper stress tensor to be associated with this. So, it was Cauchy, who gave, I believe the equation was given around 1820s. And he identified a stress tensor, which we will call T. And this is true for infinitesimal elasticity. And this stress tensor can be written as the product of 2 other tensors where, I will write down what this is, but before I do that, I just write it in component form.So, this, now I am writing in the component form, I do not have to put that. So, this is ij, so this is a stressed instead it is a second order tensor. Whereas C here is actually a fourth order tensor, so C goes from i, j, k, l and the strain tensor epsilon only takes on its again a second order tensor. So, here I will put k and l. So, this is actually C is a fourth order tensor and can account, so we can account, can account for an isotropy and we will just clarify what epsilon here is.(Refer Slide Time: 11:04)So,  here is the strain tensor. So, that is given by1 1( ) | ( )2 2T i jijj id dd dx x  =  +  = + and we will write this in the component form 2. But I will just clarify what d is, d is thedisplacement vector. d = x − X .So, I am writing it down here, this is a this. It isdisplacement vector. And these are the small x and capital X. Capital X denotes as location inthe reference configuration x refers to the current configuration and this epsilon obviouslyhere it can also be written in the component form as we have been doing before and that wewill say if it is ij then say it is given by ij  and this will be the other component here and weadd the transpose part. So, this just the indices are just flipped now, so this will now becomexi. So, this is the general form but for an isotropic material, so for an isotropic solid, theelastic constants can be reduced to only 2. So, for an isotropic solid only 2 elastic constantsare required.(Refer Slide Time: 13:15)And now and then what happens is that the form of C becomes rather straightforward. And ifI have to write it in component form, I will write it as okay this is sorry, I missed one inbetween Cijkl.( ) ijkl ij kl ik jl il jk C =  +   + Now, this becomes equal to lambda times ij  and there is another kronecker delta, but thistime that it is the other 2 indices. This is k and l will give, so we are using lambda let say μand this is ik  . And here this is going to be now the other 2 indices jl and again kroneckerdelta now you have il. So, if once you have il, what you have left with as j and k, so thesewhere these 2 are also called Lame moduli.(Refer Slide Time: 14:31)So, this has now brought us to this one was number four. So, from Hookean solid, I may likeyou can probably guess where we are going to go next right. We do want to discuss the caseof the Newtonian fluid. This will be five case of Newtonian fluid. Now, Newtonian fluid is,this is a term that you are probably very-very well familiar with. And the reason Newton'sname is associated with it is because very close to Hooke’s original work, I do not rememberthe exact year right now. But Newton proposed the concept of something called lack ofslipperiness which we understand as viscosity and from there on, obviously, the ideas weredeveloped later on.And we had Navier and Stokes who contributed heavily to the development of the equation,which you today know as the Navier Stokes equation. So, let us discuss the Newtonian fluidand it was Stokes who gave the form for the stress tensor for this particular, for the case ofthe Newtonian fluid, he was one of the first to give the proper form and he gave the form thisgoes to Stokes. So, the stress tensor here is now something called p multiplied by the identitytensor and this p is pressure basically, then you have some κ which is a material propertymultiplied by the trace of D.So, this although D is a tensor the trace of D as a number multiplied by the identity tensorplus 2 η. So, if I have to write it in component form, we have to write it as or, now by thistime you probably are very familiar with it. So, we have minus p, then you have the identitytensor, the identity tensor, it does not have to be written as I comma ij because we alreadyknow that this is given by the kronecker delta. So, I will just put ij  and then you have this κtimes this trace of D. So, the trace of D means the sum of the diagonal elements. So, we knowthat to create a sum, all you need to do is have a repeated index.So, what I can do is, I can just write it as D and then I can choose an arbitrary index becausethis is just going to be summed over. So, I can write it maybe even as ii in this particular case.And you have again the identity matrix. So, you have ij  , so since I going to write that as ij  ,maybe I will choose k here otherwise no problem. And then you have 2 η times ij  . So, wehave to now specify what D ij is, so this is a what is called as a strain rate tensor, which isgiven by the gradient of the velocity plus the gradient the transpose of that divided by 2. So,this D is also called this is the strain rate tensor.(Refer Slide Time: 19:06)Now, Stokes assumed that this κ is minus two thirds of η and so that the pure volumetricchange does not affect the stress. Furthermore, this particular term the second term here thatwe see, this κ, D, k, ij  term can be absorbed with a pressure term and that leads to moresimplified form which is now which can be written as, so I will just say simply by absorbingthe second term, we can write it as minus p plus 2 η. And, so I chose a, by the way, Iprobably want to clarify here in the Lame moduli I actually ended up using μ as a Lamemoduli. So, that is why I introduced another symbol for viscosity. I believe somewhere in thebeginning I had used μ as viscosity also.But now that we have introduced η, we just keep on using η for viscosity. Now this η is alsocalled the first coefficient of viscosity or shear viscosity and this κ is also called the secondcoefficient of viscosity or bulk viscosity. So, you can probably see the chain of thoughts isnow going to fructify. And we are going to be able to, it is going to bear fruit and we can nowintroduce some of the more intricate ideas that we want to discuss as part of this course. So,the very fact that we have introduced a Newtonian fluid should beg the question, what is anon-Newtonian fluid? So, we will just quickly write down that.(Refer Slide Time: 20:34)This was just a second, so, this was 5, so, I will say 6 here, non, so a non-Newtonian fluid is basically any fluid which does not obey the equation we just wrote. So, which does not obey, when we say it is just does not obey that means the stress tensor does not obey, I think that is easy to understand. So, if it does not obey this particular relationship, we will call it as a non-Newtonian fluid.(Refer Slide Time: 21:58)Now when you are discussing for example Navier Stokes types of fluid, we know that the viscosity is a term, but that viscosity does not actually depend on the strain rate tensor. It is independent, viscosity is a material property, it can depend on different conditions for example, temperature, but it does not depend on the strain rate tensor itself, which means thatif you keep other things constant or other things unchanged, then no matter how fast the flowis, if just loosely saying that, no matter how fast the fluid is, how fast the fluid is flowing, η isnot going to vary with respect to that. So, here we can now bring in a slightly moregeneralized idea, which is also called inelastic fluids, inelastic fluids.Now, when the phenomena dictated by viscosity only then it makes sense to model theviscosity the function accordingly. But in a more generic sense this viscosity term need not bea constant. So, inelastic fluids or generalized Newtonian fluids are fluids for which this extrastress tensor. So, if we forget for example, this part right now, then you have this particularpart, the extra stress tensor that is proportional to the strain rate tensor. But now, that there isconstant of proportionality η is allowed to depend on the strain rate itself. So, maybe I canright now, for to write down this equation form I can just say that, this is now such that I canwrite down the total stress tensor as being equal to pressure term plus 2.So, in a more general, so this η now is allowed to depend on the strain rate tensor. But what isthis. that we have written? So here, by the way, this means that η is just a function of. . Itis not being multiplied by that or maybe let me just eliminate chances of confusion where η isnot a constant. But depends on. , which is also called, which is the generalized strain rate issomething that is derived from the strain rate tensor and it is actually defined as root over of 2into the trace of D square. So,. becomes because stress is just a number, even though D is atensor your. is just a scalar quantity here.So, these inelastic fluids they have neither memory nor elasticity. So, we have achieved onecomplication over Newtonian over the very-very simple Newtonian fluids. And that is wehave said that this viscosity term can now be a function of the strain rate tensor but still theentire behavior of the fluid is governed by a viscosity term. The fluid does not have anelasticity or it does not have the ability to store energy in a sense, all the energy that you putin will start to get dissipated immediately.So, it is important to understand that is why we call it the inelastic fluid, which meanselasticity is absent. So, in a sense, you can see how we are building it up, we introduced theidea of Newtonian fluid and then we said, there is something called the inelastic fluid or thegeneralized Newtonian fluid and from here on, we can build 2 more complicated ideas. So,one of the easiest ways to understand this is through a diagram.(Refer Slide Time: 26:47)So, what we are going to try and do now is, we were going to create a plot and, on this axis, Ihave. , which we discuss as the strain rate, the generalized strain rate. And here we aregoing to plot shear. Now for a Newtonian fluid, what do you think could be the relationshipbetween the two? See, in the Newtonian fluid, the viscosity term is constant. It does notdepend on. . So, the relationship between the two is actually the shear rate is simplyproportional to. and the proportionality constant is truly a constant for all values.So, this relationship is given by line. So, when you plot this and this represents the behaviorof a Newtonian, so this is where we are going to leave it off for this today's class. And for thenext in the next class, we are also going to look at the same diagram what we are (try) goingto try and do is plot some of the behaviors of some of the other fluids for example, what wewill call the shear thickening and the shear thinning fluids, which are examples of what wejust saw, generalized Newtonian fluid or inelastic fluids. So, we are going to stop here fortoday's class and then we will carry on from where we left off. Thank you.