There are some questions on what we did in the last class. Let us clarify some of the
doubts because I know that there has been a jump in the level of the subject which we
have been dealing with till last class and lot more mathematics was involved in the last
class. So, let us just recapitulate what we did and clarify certain of the doubts.
The first confusion that usually students have is what is the difference between stress
vector and stress tensor? The first thing that we should understand is that tensor is
a generic name, generic name. Tensor can be zeroth order. Zeroth order tensor is nothing
but a scalar.
The first order tensor is called as the vector. Then the tensor keeps moving to second order,
third order, fourth and so on. It is a generic name. Now, the question is what is a vector
and what is the stress vector and what is the stress tensor? If you remember we were
dealing initially for the definition of stress, a vector quantity; for the definition of the
stress, we were dealing with the vector quantity. All of you know that vector has a magnitude
and direction. It is at a very fundamental level or I would say in the earlier classes
you would have defined vector to have something magnitude and direction.
For example we took that small surface around the point P and then said that let this be
the normal of the surface and let this be the force or the resulting stress vector.
This is nothing but let me call this is as delta f or F which means it is the force and
remember that we said limit as delta S tends to zero, we define stress as delta F by delta
S or this is nothing but the definition of df orF by dS. I need not even specify that
it is nothing but df or F by dS and this is what we called as a vector T. Because this
is a vector, we divided this by an area and hence T is a vector.
Now, we put one n there and a squiggle at the bottom. This is to indicate that this
quantity is a vector. This is to indicate that the T depends upon the plane that you
choose; we saw that already yesterday. Your question is what do you mean by T1 n, T2 n
and T3 n? That is the question. What do you mean by T1 n, T2 n and T3 n? What are these
things? Is that right? They are nothing but the quantities associated when I resolve this
vector in the x1, x2, and x3 directions. This you would have In your earlier classes you
would have written T is equal to T1 ni plus T2 nj and T3 nk plus. In this course we are
not going to use i j and k because of the indical notations that I am going to introduce
shortly. Hence instead of i j and k I call them as e1, e2 and e3. They are called as
the basis vector. So, T is now defined as T1 e1 plus T2 e2 plus T3 e3. So, this is a
vector. We went ahead and derived a matrix ultimately for sigma and called this as state
of stress at a point. Remember that we called this as a state of stress at a point. We said
that there are 9 quantities, 9 quantities with it.
We associated two indices with the sigma i and j. One index, the i index indicated the
plane, j direction. So, i and j are two indices whereas here, how many index? Only one index.
The 9 quantities cannot be expressed in terms of three baseses as e1, e2 and e3. There have
to be nine baseses to express this. Let us not worry about too much of theory in tensors
right now, but remember that this has 9 and that has 3. So, this is called as a second
order tensor. People deal with this in a matrix form. That is an easier way for an engineer
to deal with a second order tensor. That is an easier way to deal with the second order
tensor. But though mathematically there is a difference between viewing it as a matrix,
viewing it as a tensor, but for all practical purposes when we do manipulations with it,
the concept of matrix is extremely useful to us. Hence we look at it as a matrix. Why
did we define this? We had a long discussion and ultimately we came up with a relationship
between the sigma and T.
You remember what was their relationship? We said that say for example Ti n is equal
to sigmaji nj. So, we came up with the relationship of this kind. What does this relationship
tell us? It tells us that if you give me a state of stress at a point P which means that
you give me all the sigmas, sigma11, sigma22 and so on and if you ask me a question what
is the stress vector at this point P at a plane n, then using this equation I will tell
you or I can calculate and tell you what are the T1’s, T2 and T3 or what are the Ti's
rather? I will be able to tell you Ti's. That is what I mean by this expression. Now, let
us expand this. Is that question clear? Is that clarifies your doubt? Please also note
there is one more question that what is T and what is n? They are totally different
quantities or what is delta f and what is n?
See, sometimes when we talk and when we use it we have a tendency to say force and then
say corresponding stress; stress vector and the force vector. I think when we use it,
it will become very clear the difference between the two and this definition here will tell
us what is the difference? Please note that both of them are different. One is the normal,
the other is the force that is acting. There is a force that is acting or the effect of
the other body onto this body where P sits. If I consider a corresponding point in the
other half of the body, remember I had cut it into two pieces; corresponding point in
the other half of the body, then the force will be acting in the opposite direction,
like bar we ….. I do not want to repeat what we did, but like that bar where we cut;
one side there was a force like this, the other part will have a force in the opposite
direction.
What does this equation now mean? There are indices. As I told you they are summation
indices. There are rules for summation. These indices have in other words These rules are
not based on lot of physics; you know it is based on certain rules to make things easier
for us. Why do you sum up? You cannot ask a question because it is a convention; it
is rather a convention like for example you write a vector with the bold letter, like
that it is a convention. What is the convention? The convention says that any vector or a tensor
or whatever quantity it is, can be expressed or even a scalar can be expressed in terms
of a letter and an index associated with it; by means of a letter and an index associated
with it.
For example, v is a vector. All of us know that it can be resolved into v1, v2 and v3
and we express this as vi. Sigma, what we just now saw? We just now saw sigma and we
associated with sigma two indices which we called as sigmaij. I can associate, say for
example, what is called as elasticity tensor, say C with the index Cijkl. These indices
are not repeated. i and j are different and so they are free indices. At one level, at
a very simple level, this also indicates the order of the tensor; number of free indices
indicates the order of the tensor.
I can write say for example p as sigmaii by 3. There is a difference between this and
this? What is the difference? What is the difference between the two? It gets repeated,
i now gets repeated. Here it does not get repeated, here it gets repeated. The convention
here is whenever an index gets repeated it means that you have to sum up over the range
of that index which means that sigmaii is equal to, what is it? Sigma11 plus sigma22
plus sigma33.
So, sigmaii is equal to sigma11 plus sigma22 plus sigma33. Here we have divided by three;
it does not matter. Sigmaii means that sigma11 plus sigma22 plus sigma33 whereas, here in
this case when I say about sigmaij there is no repetition, which means that i and j takes
values ranging from 1 to 3. So, I have 9 different quantities. That is what it means. This means
that I have Sigmaij means that I have 9 different quantities whereas when I say sigmaii, it
means that there is only one quantity. What is this sigmaii then? Whether it is a scalar
or a vector or a tensor or a second order tensor or what is that? Yeah! That is true
that is scalar because there is only one quantity. There is only one quantity and hence this
is a scalar. So, this kind of index which does not get repeated, stands out free, tells
me the order of the tensor. So, this is a fourth order tensor, this fellow is the second
order and so on.
Tensors are also classified by transformation of coordinates. I will not deal with it in
this class, I will take it up in a later class. The tensors also are dealt with from transformation.
We will go step by step to understand it, let us not lose the thread. We will come to
that later, but right now let us only worry about how to study or how to understand the
indical notation; I know that they are difficult notations. It is not that always we put these
indices as a subscript or below here.
Under certain conditions, we can also or we will also put it as a superscript as v i.
But, in this course fortunately we will not use indices as a superscript but it has its
own meaning and so on. Now we will only use it as a subscript in this course. When I go
to superscript, then I have to deal much more with tensors which I am not going to do, but
we will deal only with this kind of indices. Summation index, as you right now would have
understood and got a grip of it, is that whenever there is a repetition you have to sum up.
Again, this mechanics people have a convention.
They say that as long as you use roman letters for subscripts say i, j, k, a, b, c, d, whatever
it is, these roman letters then the range of the subscript is from 1 to 3. So, when
I say sigmaii, my range is from 1 to 3. You can also use for subscript what are called
as Greek letters. So, say for example I can use alpha, beta, gamma and so on. I can use
for example sigma alpha alpha, subscript; instead of i and j, I can use alpha, beta,
gamma and so on. So, when I use alpha, beta, gamma, my range is restricted to 2 or in other
words my range is 1 to 2. What does that mean? It means that sigma alpha alpha means that
it is sigma11 plus sigma22; that is all, sigma33 does not come into picture. Only when I want
3, the range to be 3, then I use i j and k. Any questions on it? Now let us go back and
look at this expression. Having clarified all these things, let us now write down this
expression.
Look at the left hand side and look at the right hand side. What do you see in the left
hand side? You see one i. On the right hand side you see i stand out or free and j is
what is repeated. So, what do I do now? I sum up over j. This expression can be written
as sigma1i n1 plus sigma2i n2 plus sigma3i n3. This
is for one i, where or 1 expression say i is equal to 1; rather I should say when i
is equal to 1, I get one equation. Then, when i is equal to 2, I get another equation. When
i is equal to 3, I get three equation. The three equations with each of the equation
having three quantities are summed up into one simple expression. Please note that this
j is a dummy index. Because of the statement which we made just now, I can replace j by
k, for example.
I can replace this by k. Do my expressions change? No; it does not matter. But when I
replace it by alpha, beta it will change because I will lose one term. This is a simple summation
convention. In your earlier classes what you would have done? You would have put one sigma
before it and you would have said that by putting a sigma and i varying from 1 to 3
and j varying from 1 to 3 or k varying 1 to 3 this is what you would have done.
Now instead of doing that what is that I have done? I have said do not worry about the sigma,
putting a sigma here; do not do that. But whenever you get a repeated index you sum
it up. Note that your left hand side and right hand side should balance as far as these indices
are concerned. What do I mean by that?
Let me write one more small expression which we will use later to understand what I am
saying. Let us write Cijkl epsilonkl. Just take a second to understand what this means.
Look at that. You have ijkl kl? What is this first one? It is a tensor, correct. But what
is the order? Fourth order. What is this? Second order tensor; so, fourth order tensor
now operates on a second order tensor. What it means? It means that this is something
like a function which operates.
You would have probably seen and understood very well say y is equal to some function
of x, more sophisticated fashion you would have called this function as a mapping, function
maps x to y. You would have written it in a different fashion. As you grow and as you
take higher and higher levels of mathematics, initially you would have called this is a
function, then you would have called it as a mapping and so on. So, you can say that
this is an operator. It operates on epsilonkl to give one more quantity. Please note that
this quantity has to be what? Correct; it has to be a second order tensor so that my
indices balance each other. In fact this is the famous expression for the stress strain
relationship.
See, sigmaij; that ij fellow is here, then kl is here and epsilonkl. So, ijkl epsilonkl;
kl gets repeated. So I will have how many equations by the way? Totally how many equations?
9 equations, correct. i and j 11, 12, 13, 21, 22 and so on. So, I can have 9 equations
and each equation will have how many terms? 9 terms, correct. Because kl will have values
ranging from 1 to 3, so I will have 9 terms. Totally I have to write 81 terms; 81 terms.
What is that I have done? I have summarized the whole thing in one simple single equation.
In fact, as you go along many books, they will not even use this kind of indical notation.
What they will do is to write this as sigma is equal to C epsilon. Some books put two
colons or colon rather, two dots before it and so on. That is why, say, this convention
how to write it may vary from one book to the other. So, you have to look for it. Actually
these are tensor operations. We will not go into the details of it, but it is important
to understand what these indical notations mean. Any questions?
Yeah! There can be one small doubt, ….. what happens when this kl gets repeated 3 times?
2 times is fine, why not 3 times? Can I have one more term and have again another kl? In
tensor analysis usually it is a bad practice, again I am saying it; it is a bad practice
to write thrice. It should not get repeated thrice. But many times or a few times, I should
not say many times, sorry for that; but few times it may so happen that you may have to
write it again. It may so happen, in which case the convention is that we will abandon
summation; summation is not implied. As far as possible you should not write it, repeating
it three times. You should write it such that it only repeats twice. Now, having understood
this, we will move to slightly more difficult conventions.
These conventions come from two letters called delta written as deltaij, which is called
as Kroneker delta and a permutation symbol, epsilonijk; this is called as permutation symbol. These two
symbols have some special meaning to it; these two symbols have some special meaning to it.
Let us understand what these things mean? Now, the first question that you may ask is
what is so great about these symbols or why is that these symbols are used? Why do you
want to introduce this? These symbols are used for easy manipulation, as you will see
in a minute. First let us state this symbol; it is again a convention.
The convention states that when i is equal to j, how many quantities are there by the
way for deltaij? 9 quantities; so, again it is a matrix, delta11 and delta12 and so on.
When i and j are the same which means that it is the diagonal of the matrix delta11 delta22
and delta33, then this Kroneker delta takes a value of 1; takes the value of 1, when i
is equal to j. It doesn’t mean ii. Please note, it is not i i. Deltaii has a different
meaning, that is summation. When i is equal to j, that means 11, 22 and 33, then deltaij
takes a value equal to 1. When i is not equal to j, Kroneker delta takes a value of zero,
when i not equal to j.
Can you recognize what this really means? Can you put down for a second this in a matrix
form and tell me? Correct; unit matrix. So, when I look at it, it is 100 010 001. In the
matrix notation this becomes a unit matrix.
We will look at small operations with this in a minute. Before that let us understand
what is epsilonijk? Epsilonijk is a permutation symbol, can take on values with i j k ranging
from 1 to 3. So, it is a third order tensor. This is a second order and that is a third
order tensor. As the name indicates this is a permutation symbol and hence we can write
first 1 2 3 in a particular order, say. Whenever ijk is such that they take a clockwise order
that means epsilon123, 123 312 or 231, we say that the corresponding value of epsilon
is equal to 1. These are clockwise.
This is for clockwise 1 2 and 3. Whenever I have anticlockwise, instead of 123, 213,
321, and so on, this permutation symbol is deemed to have a value. It is just an assumed
value. We say that this will have minus 1 for anticlockwise. Otherwise, whenever I get
this index to be repeated, for example epsilonijk becomes 112, it is a part of it; 112 or 122
or 121 and so on, whenever this index gets repeated it is supposed to have a value of
zero.
We will do two simple problems to understand, where how we use this. Before we go further
we will just try to understand. Is there any question on indical notations? Any question?
Now, I will give you a problem in just two minutes so that just think about it and we
will discuss it in a minute. I hope these questions are clarified. Are all these things
okay? Whatever we have done so far okay? Yeah! I know there are one or two more doubts, I
will clarify it towards the end of the class. We will just follow what we are doing now.
Some doubts of what I have done in the previous classes I will clarify it towards the end
of this class.
Now the first question is what is sigmaij deltaij? Sigmaij deltaij, what is this? Sigmaii,
is it diagonal matrix? No, it is a scalar, very good. So, it is sigma11 plus sigma22
plus sigma33. What is that you do? Simple; when I look at i and j for a delta, immediately
I know that when i is not equal to j, I know it is zero. So, the only terms which will
now be available for me to sum up are terms where i and j are the same. I just go ahead
and put for this indices i and j to be the same, in which case sigmaij deltaij will become
sigmaii; so, sigmaii.
I will give you a tougher problem, slightly; wait for a minute. Let us say C is a vector
and that is formed by our famous cross product. I want to now write How many quantities are
there? C1, C2 and C3 or Ci; similarly a, similarly b. I want to write this in terms of C, rather
Ci’s, in terms of a’s and b’s. Let me not use ai’s and bi’s. Let us say a’s
and b’s; a1 a2 a3, b1 b2 b3. Can you do this by using epsilon? That is the question.
Can you do that using epsilon? What you do is, write it down as you know in terms of
C1 in terms of a1 a2 a3, b1 b2 b3 and see whether you can introduce now epsilon. If
you have questions, you want time, do you want a minute or want me to tell the answer?
Just try it out for a minute.
Let me give you a clue; Ci. Just verify; Ci can be written as epsilonijk aj bk. Ci is
equal to epsilonijk aj bk. Note the repeated indices j and k. What happens to C1? You write
it down in terms of j and k. Just check up whether we will get the same result. Yes,
so how did you get this value first? One is note that Ci I will keep this as 1; so, C1
sorry epsilon1, when I say when I put 11 what will happen to this? Zero. Then 112 again
0; 113 again 0; so, the one, the first fellow who will have the value here is 123; 123 which
will have a value of 1. So please expand this and check up whether you get the value, the
cross product that you know of. I am not going to do it because it is very simple. As an
exercise, you please do that and now you will understand that the use of epsilonijk is also
there to easily write down expressions like this. Such expressions can be very easily
written down and you will have more and more use of these quantities later in the course.
Any questions?
Having learnt now the indical notation to certain extent, we will go to study what is
called as equilibrium equations; we will study equilibrium equations. All of you know equilibrium
equations, but what I am going to do is to express it more rigorously; I am going to
express this more rigorously. What is an equilibrium equation? For a body which does not deform,
immediately you will say sigma F is equal to, sigma F what is that? Zero. Now sigma
M is also equal to zero. One of the questions that can come to your mind is why is that
I am always putting delta F and then looking at it as a sigma?
Because for an equilibrium sigma F is equal to zero, sigma M is equal to zero; moment
also should be equal to zero. You remember that when I had cut this body by means of
a plane, remember we did that, and I had a point P and I had put only F there. I had
not put M there. In normal continuum mechanics and in solid mechanics, we do not consider
M, moment. When you want to consider moment also, it gives rise to what is called as couple
stresses, which we are not considering in this course; we are not considering that in
this course. So, let us not worry about … When I cut it, it does not mean that the body is
under equilibrium even when sigma M is not equal to zero, it does not mean that. When
I cut it and look at the influence, we assume that M is equal to zero at that point and
we do not consider M at all. So, let us now first consider what happens to a body when
sigma F equal to zero.
Let us put the body. That is the body of interest to us and the body has two types of forces
that are acting. What are the two types of forces, body forces? Let us say that is the
volume inside, sitting inside dv and that there is a body force that is acting and let
me call this body force as say F. Please note, it is a vector and there is also a surface
force that may act on the surface, say dS and that surface force we can call it as,
say T. What is that I have left out? n? 4240Now, the first thing that I am going to do is to
write down an equation which all of you know, sum of these forces is equal to zero. How
do I write that down? I will first integrate all T’s, surface forces. So, it is a surface
integral designated by S T n dS.
Please note that it comes directly from my definition. So, T n dS or rather I should
say that it is a vector, I should I mean sorry stress. I should not call this; always I have
a tendency to mix force and stress. So, I hope when you are following it up, we follow
this as a or a traction; I should call this as traction and a stress vector that is acting.
So, this would be the total force. I am sorry for that; this is the total force. Whenever
I put T and say force what I mean is actually the stress that is associated. The context
will tell you what I really mean. This plus what is the other thing? Body forces, where
I have to take a volume integral. Always the students have a tendency to get scared about,
you know volume integral and surface integral and so on, but what I am going to deal in
this course is only the concept. Do not worry that you are going to do lot of integration
and you have to study this lot more and so on. I am going to only look at it conceptually.
With respect to volume, how do I write this? Fdv is equal to zero. That is the equilibrium
equation. It is possible to write this down in an indical notation. Though I would not
like to do it quite often, but it is possible to write this down in an indical notation
as well just for clarity because we are just starting on indical notation. Please note
that this equation is valid for or I can resolve T and F in three direction, valid for all
the three directions.
As a common factor for three directions, I can write it as, in terms of i as integral
S Ti, i will take values from 1 to 3, n dS plus volume integral Fi dV is equal to zero
or in other words T1 and so on. What is Ti n? Sigmaji nj; so, I can write this as sigmaji
nj dS plus integral V Fi dV is equal to zero. At this stage, I see that one is the surface
integral and the other is a volume integral. Now I want to convert surface integral into
a volume integral. How do I do that? Divergence theorem; let us not worry about the intricacies
of divergence theorem or how it is derived and so on. We will just write down the result
of the divergence theorem; divergence theorem is used to convert.
Divergence theorem is used to convert the surface integral into a volume integral. Yes,
I know some of you might have forgotten divergence theorem, so, I suggest that you just have
a look at it and come back. We will continue this derivation from the place where we left
in the next class. But, before we close I want to clarify one or two things and introduce
one small notation in order to write down the divergence theorem.
It is possible that we differentiate some of these quantities. Sigmaji or ij can be
differentiated with respect to say x1, partial differentiation with respect to the x1. So,
I can write this as a dow by say dow x1 or x2 and x3. In other words I may differentiate
this with respect to i and so on and then sum it up.
Whenever there is a differentiation like this, I write this down as dow sigmaij comma, sorry
sigmaij, comma j or comma i or comma k which means the order will increase now. Please
note that depending upon what I write here, the order will increase or decrease. This
comma indicates that I have differentiated it with respect to xj and j gets repeated.
We will use that for divergence theorem in the next class and maybe we will start the
class with one or two clarifications which our friends have sought on what I have done
before. I did not want to do it because I did not want to stop the flow of the stresses,
that concept of stress that we are dealing with and hence I just carried on with it.
We will come back and recapitulate what all we did in this class and the previous classes.
We do not lose track of it and go over to further definitions. If there are any questions,
we will start with those questions in the next class.