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Complex Variable Representation

So, welcome back to one more lecture on introduction to Soft Matter. This is as we said, this is towards the end of our lecture series and the last chapter that we are discussing as part of this course, is the issue of sinusoidal inputs and the output, the consequent output from a viscoelastic material. And last class we saw that when you input a sinusoidal strain history, you get an output which is also a sine function, but which has now a phase lag and a phase difference there exists a phase difference between the sine between the input and the output, right. And all that where you are using regular trigonometric and there is also another way similar but very useful format in which we can put the same equations, which is a complex variable representation. So, we will just do that today. (Refer Slide Time: 01:22) So, we want to look at the complex variable representation. So, recall cos sin ie i where i = −1 . So, we can represent our input in terms of a complex variable using this. So, how will we do that? We will use, so now, our input is epsilon s is epsilon naught, instead of writing a sine function, we will just write the complex variable representation which is and s is again a real number between 0 and infinity. Now, if so, we had discussed what the, if the inputs are cosine and a sine function, their responses we discussed in the last class. So, let us just write down, if you have 0 ) then it leads to a response and the response now is some functions of time, which we did last time. And if you have the response as a cosine function, sorry this is, let us write s for sine function and let us write c for cosine that will help us distinguish. Then the response is some , this is an s, s for sine c for cosine. So, now if we use the result that we just derived in the previous class then we can say that what is the expression So, what we want to do now is let us get all the imaginate the sine terms together sorry G dash terms together. So, what we should do now is we should get all the similar terms together, so we will get 0 So, what I am going to do is I am going to try and represent this itself as a complex variable. So, see what we are trying to do? Our input was a complex function, we are trying to reduce this also to a complex function. A complex function that is easy to understand this is already in a complex format. But something that might be easier to, so we have the first term which we will leave it as it is. And then the second term, I am going to bring the i out. So, I will just put i and this is G'() . This is now going to be cos(t) +i sin(t) . But what is now you see can see that both of these two terms have the same time varying part, this is equal to i t e and similarly, this part is equal to i t e . So, I can simplify my case further by writing epsilon naught, this is now and this is a very important number, this is called the complex modulus and G is called the storage modulus and the G'' is also called the loss modulus (Refer Slide Time: 10:05) This complex modulus is now another important material response function which means that G' can be used to characterize the behavior of a material. Earlier we had when we did not have sinusoidal inputs when we just had a step stress or a step strain we saw that there were two important functions that were material responses and that was the stress relaxation function G(t) and the creep compliance J, right J(t). And now, we see that when we apply a sinusoidal strain or a sinusoidal input, when we put it in, when we express the sinusoidal input as a complex number, we can get, we can characterize the output in terms of another complex number, where you now encounter this new quantity which we are going to call the complex modulus, right. Now, just in terms of the previous variables, so recall we had written before . So, see this is giving an important equation right here, because it is tying your complex modulus with the response that you have determined before right. G(t) is a response that you can determine separately and if you can rewrite it as a constant term plus a time varying function, then you can this is the method by which you can compute the complex modulus analytically from that. So, obviously here we imply that the functional forms o , this must be known to us. Now, we will go on with this writing this entire thing in this in the form of i e (Refer Slide Time: 15:24) So, we can make some more simplifications, some more important equations we can do. So, this G*() , this is now obviously a complex number. So, this complex number can again be rewritten as the amplitude of this complex number multiplied by e to the power of something. So, these are the different forms that you will probably encounter in literature at different points of time. If you go through different books or different materials, then these are usually the same idea is put in very different forms and these are some of the forms that you are veryvery likely to encounter. So, now this tells us that your stress is going to be, so this was, so, what is going to be the (Refer Slide Time: 17:50) Another term that you are going to you are very likely to find in literature is the term called complex viscosity and this is very much related to the equations that we just derived. So, let us use the idea of the Kelvin voigt model. So, in the Kelvin voigt model for a Kelvin Meyer voigt body your equation was . So, given the complex form, what can we do about the stress? So, let us say, so in this particular case your stress is now going to be ] , I am just rewriting the previous equation for . So, we have already seen that. Now 0 . So, this implies that my stress is, so this first part is epsilon and the second part. So, I recast that equation. This is the generic response, right. And I have recast this for the case of the Kelvin Meyer voigt body equation. So, this is the governing equation for this and using the general response, , this term is called a complex viscosity. So, we use this as a motivation. So, today what we have done is we have seen a number of different formulations or quantities that you are very-very likely to encounter when you are reading literature on viscoelasticity. And one of the important terms that we today discuss was *G which is a complex modulus and we saw that this broken up into two other moduli, which is called a storage modulus and the loss moduli. Now, the two names are obviously given because one of them represents the storage or the storage of energy in the system. The other one represents the dissipation of energy. And this idea of storage and dissipation goes back to our previous discussion on the classical elastic bodies, the classical elastic solid and the classical viscous fluid and from there on, we have motivated ourselves and seen that these two that one of those terms is a represents a storage and the other one represents a dissipation, right. Okay so, having derived all this, there is one example that we are going to start and I will encourage you to try and finish it in the next class and we will, that sample problem we will finish next time but I am going to start the problem today. So, let us take the case of Maxwell fluid, the governing equation for this we already know is some 1 p , we are recasting it now that we know that both the sides are basically polynomials in the operator D. So, we are just rewriting it as in that particular form. So, 1 p has a certain value that you can find out going back to the notes and so, with the first derivative I write 1 p , the term which does not have a derivative at 0 p . And on the other side, we have the derivative of epsilon. So, I write 1 q . And for this you had found out that the function G(t) had a particular form, it was actually − where lambda was the characteristic time scale associated with this also called the relaxation time scale. So, for the case of the Maxwell fluid, we have, so we are trying to go back and see reevaluate some of these moduli in terms of for the Maxwell case. So, you know that in this case is 0. So, and that means that the time dependent term is simply this one. (Refer Slide Time: 25:29) So, what I am asking you to find is, for example, what is the value of the complex modulus? And we know already the equation. So, I just want you to compute the integral. We have, we know that the general solution for this is given as So, you want to replace this, the first time is actually going to cancel. It is just becoming going to become 0 and then you replace this term here and you try to see what result you finally get. So, today, this is the second last lecture. And what we are trying to do right now, this is what we are finishing up the chapter on sinusoidal response to sinusoidal inputs. And what we found is, we found ourselves, we familiarize ourselves with some other very important variables that come again and again in viscoelastic literature, which has a complex moduli, loss modulus, storage modulus and also complex viscosity. And towards the end, we as this is the case of the Maxwell fluid and what we are trying to do is to find the functional form for the complex modulus case. So, I left I leave you to here, this is almost done. So, for the next class, I would encourage you to go ahead and solve this and we are going to discuss the result. And we are also going to wrap up in the last lecture. So, today we will stop here.