In the last class we were enquiring about
a quantity called stress. We said that stress is the result of forces. Remember that we
said the forces can be classified into what are called as surface forces and body forces.
That is where we stopped. We started defining what is stress? What was the difficulty we
had? We said that stress, in our earlier classes rather that stress is nothing but force per
unit area. ….. divided this or normalize the force by area and it was very simple.
But when the body takes the shape of a very complicated connecting rod or something else
like what we saw in many of our examples or for example the sheet of the side wall of
a coach and so on, then it may not be very easy for us to define or to use this simple
formula. It is okay as far as a bar is concerned; as far as the bar is concerned. If I apply a load like this, then, yes, it
is quite simple to state that the stress can be defined as force per unit area or sigma
is equal to F by A. But the problems or the real life problems rather are not as simple
as this. So, we have to have a much more rigorous understanding of stress. Let us see the definition
of a stress from a more rigorous three dimensional setting. Now let us consider a body. Let us put some coordinate system for it.
Let us call this as X1, X2 and X3 and let us say that there are forces that are acting
on this body and that there are some, what are called as boundary condition. Remember,
we had already talked about the importance of the boundary conditions in one of our earlier
classes. Let us take this body out and let us cut this body by an imaginary plane. Our
interest is to define a stress at say a point P. Why is that we are interested in a point
P? Remember, when you saw our results, you saw lot of colors there. So in other words
stresses were defined throughout the volume of the body. Hence we have to define stress
at each and every point of the body. So, let me take a point P and let me cut the body
with a plane which passes through this point P. Let us say that the normal of this plane
is equal to n; all of know that n is a vector. That is the normal. Let me know keep these
two body separately. Forget for a moment all these forces and the boundary conditions and
let us consider the two bodies separately. My point P is here. Let us say that we take
an area around this point P and call this as say dS. Remember that the normal at this
point is n. The normal at this point is equal to n. It is obvious that if I call this as
say one part of the body and the other as the second part of the body, then there is
an influence of the second part of the body onto the first part of the body. The influence
is through a force. Let us say that the influence at this point P is equal to some F or if you
want you can say delta F. That is a force that is acting. Now, let us stick to our old
definition of stress, but the only difference now is that we are going to talk about area,
which are around a point P or let us now say that the normal, the average of these forces
that act at this point P can be given as say delta F by say delta S; for a moment to stick
to our delta F here let me call this as delta S as well. The average force or the average force is
due to the influence of the other body at an area surrounding the point P is given by
delta F by delta S. When I started this problem, I said that let me take a small area around
P and now what I have done is to take an average of delta F by delta S. There are two questions
that may come to our mind. One is that how small is small, because as I take larger and
larger areas, the force distribution may not be very uniform. So, I will get different
values of this average. Remember, this is force, area. The definition is still there;
our old definition is still there. So, one argument is that if I keep taking different
areas around this, because you have not defined how small is small, so, I can take one area,
then another area and so on I may get different values at this place. Because I am averaging
out in an area and especially when the force distribution around this point P as I cut
it and look at the influence of this body onto this body the force distribution may
not be very uniform. If it is uniform there is no problem. If it is uniform then delta
F by delta S would remain the same. But, if it is not uniform, the average may rise up,
go down or whatever it is. So, I have a small problem there. In order to alleviate that problem, I define
a stress as limit as delta S tends zero delta F by delta S. This is what I define as a stress
say T. Since I have got this stress by cutting with a plane whose normal is n, I call this
as Tn, with a subscript n. We will come to this in a minute as to why this n becomes
important physically and then more mathematically. We will come to that in a minute, but before
that let us look at this quantity. Please note that this quantity is a vector. It is
a force. The influence of this body over this at that point which is of interest to me.
In other words this is a force, influence comes out as a force. This quantity is a vector
because this is nothing but an area. So, this is a vector. We call this T with such a definition as
stress vector. If you remember, if I had this particular bar and in fact if I cut this piece,
if I cut it into two pieces like 1 and 2 if I had asked you to draw a free body diagram
for this, you would have drawn in this side something like that and this side a corresponding
force that is acting in the opposite direction and you would have told me that this is nothing
but the internal force which is the result of these stresses. You would have said that
this part is sigma A, that part is sigma A and you would have said that this is under
tensile load. Just extend that here. The only difference now is that as soon as I say what
the stress is here, you had immediately a plane in mind which is 90 degrees to the axis
of the bar and we would have taken this plane here. But here what is the stress of this
point? Then I have to specify the plane. Now, physically what does it mean? Physically what
does it mean? We have defined a stress at this point as
say F by A where F is the force that is acting and A is the area and we had actually looked
at the equilibrium with respect to the outside and the internal forces. Suppose I had taken
a plane like this, what would be the stress with respect to that plane? With respect to
this plane if I cut a body like that what would be? It would be zero. Or in other words
at this plane the stress would be zero whereas on this plane the stress would because F by
A or in other words physically we know that stress would depend upon the plane. How exactly
does it depend upon we will see that in the course of this class but it is very clear
that the plane becomes important and that poses a problem. What is the problem? This
definition of stress vector is for a particular plane whose normal is n. So, suppose I ask
you a question what is the stress at a point P, at this point P which can be related to
my point in the actual component? Then the question that you will ask me is what is the
plane that you are talking about? Going back to this picture here, I had chosen a plane
n. The question you will ask is whether you want
the stress at this point P along this plane or along this plane or along this plane or
along this plane and so on or in other words this stress at a point P becomes meaningless
because that is also wedded to the normal that you can draw or the plane which you use
to cut it into two pieces and look at the interaction at a small area around P. So,
infinite number of planes I can cut. There are infinite numbers of values of stress that
you would talk about or infinite values of stress vectors that you would start talking
about. That is quite confusing, isn’t it; that I cannot talk about infinite number of
stresses. What is the remedy? One thing is clear that the stress depends upon the plane;
another thing that is clear is that I cannot just talk about stress at a point P. So, the solution to this is to talk about
what is called as state of stress at a point at P. What is state of stress at a point P?
From this definition it is clear that if I am able to define certain planes very nicely,
then I can say that state of stress is related to certain well defined planes or in other
words what is a plane which is easy to define for you and me? Yeah, so, that plane is defined
by my coordinate systems. For example in the same figure if you want the state of stress
at a point P on the same component, we want the state of stress at a point P, then I can
define three planes, 1, 2 and another plane perpendicular to the axis of the board such
that the normals of these or in other words, the normals of these planes pass through x1
x2 and x3. These three planes are well defined planes
and the first thing I will do is I know that it is difficult to define infinite number
of planes. So, the first step is let me define them in these three planes. It turns out as
we will see that these three planes are useful, very useful to us because if I properly define
stresses in these three planes as I am going to do now, then it is possible to relate these
stresses to any other plane that you would define by means of a normal n. If it is possible for me to say what is the
stress in this plane, I can derive a formula to say if you give me what is n then I would
tell you what would be the stress in this plane? Now, let us go ahead and define the
stresses in these planes. Remember that right now we are talking about stress as a vector;
it is stress vector. It has both magnitude and direction. Let me take this point P and then expand what
happens in these three planes. Let me take that P here and let me draw a plane whose
normal is say along x1. Let me call the base vectors along x1 x2 and x3 as e1 e2 and e3.
That is x1 x2 and x3 each have base vectors as e1 e2 and e3. If it is not visible let
me write it quite clearly. For x1 we have e1, x2 I have e2 and along x3, I have e3.
Note that I put a squiggle here in order to denote that they are vectors. We will put
an arrow at the top on its head like this or in text books you would see that this letter
will be written as a bold letter. It not easy for me to write a bold letter here, so, I
am putting a squiggle at the bottom to denote that they are vectors. e1 e2 e3, in your earlier classes, you would
have been exposed to this. In your earlier classes you would have defined e1 e2 e3 as
i j k; that is right i j and k. Because of certain reasons which will become clear maybe
towards the end of this class, I have defined them as e1 e2 and e3. Now let me say that
this force that is acting on this particular plane is equal to say delta T. How do I put
here? e1 and the corresponding stress due to this as T e1. What is this? It is exactly
the same as this. Here I have taken some plane with a normal n and defined this stress vector.
In this case, I have taken a plane whose normal coincides with that of e1, along with the
same direction e1, and then I have defined a stress. I do the same thing with respect to this point
say P; we are considering that point is P. I am doing the same thing or I am going to
do the same thing with respect to the point P with the plane taken such that its normal
is along x2 in which case I would get one more, please note this is different, one more
stress what? vector, which I would write as e2 and lastly I will take another plane. How
do I take another plane? Very good; so, the normal with normal in the direction of x3,
which I would call as whose unit vector is called as e3. So, I will get one more stress
vector which I will call as e3. So, I have defined three stress vectors. As soon as you
see a vector what is that which comes to your mind? You say that this vector can be resolved
in three directions. I can resolve this vector in three directions so that I can write this
as, how would do I write? T1 in the 1 direction e1 into how do I get?
Correct; e1 that is it, plus T2 of e1 into e2 plus T3 of e1 into e3. This vector is resolved
in the 1 direction, 2 direction, 3 direction. Note this very carefully that this vector
has e1 as the plane. So e1 e1 and e1 remains the same and this 1 2 and 3 indicates the
directions and e1 e2 e3 are the unit vectors along the three x1 x2 and x3. Similarly I
can write this as T1 e2 into e1 plus T2 e2 into e2 plus T3 e2 into e3. You can write the last …. again as T1 e3 into
e1 plus T2 e3 into e2 plus T3 e3 into e3. Already I know that you are complaining too
much to write; I wantedly wrote this because I know it is every time you write like this
it is going to be difficult. So, you would immediately request me to write it in a short
hand. I mean, you have to write so many things, it becomes difficult. We will device a short
hand technique to write such a complicated equation or so many numbers, which has so
many indices and so on. We will device a very simple method of writing it in a minute. It is usual or it is customary to write T’s
in terms of what is called as sigma. Sigmaji with T such that j indicates the plane and
i indicate the direction; j indicates the plane and i indicate the direction. To understand
this if I put Te11 that is in other words what is this? Sigma11; this is sigma12 and
this is sigma13 and so on. See, some of the books would define this as sigmaij, a few
of them. For example Kolsky, very famous book on Stress waves in solids, he would define
this as sigmaij, i is the direction and j is the plane. But it does not matter because
you would shortly see that this sigma is symmetric or ij is equal to ji. But, usually it is the
plane and the direction, which are the ones that are used to define sigma. Yeah, any questions? Yes, now the point is
that I have now how many quantities? 9 quantities, 9 quantities to define state of stress at
a point. When I show you sigmaij is equal to sigmaji, then it will reduce to 6 quantities
but, as such you can see that we have 9 quantities. So this sigmaij is defined as the state of
stress at a point. So, state of stress at a point is defined
by means of sigmaij and which you can say is matrix of sigma. They are nothing but sigma11,
sigma12, sigma13. So these 9 quantities are the crux or the clue to stresses at this point.
Please note that we started with stress vector and resolved that stress vector in the three
directions, do you remember? That is what we did. The question is then, what is this
sigma? Is it a vector? Is this stress, state of stress defined as a vector or is it something
else? Can you call this as a vector? No; obviously no, because from the vector, stress vector,
quantity only we have come to this and also how many quantities are there? 9 quantities. So, this sigma state of stress is not a vector
but what is called as tensor. To be precise it is a second order tensor; it is a second
order tensor. Let us not right now worry about this word tensor; you need not be too much
taken aback by sudden jump in mathematics, I will define tensor if necessary later in
course more precisely; what is tensor and so on but nevertheless you should understand
that stress can be expressed as a vector or a tensor. So, you cannot say, see for example,
certain things like for example speed; what is speed? When I say speed, it is scalar.
When you say velocity, you would define this as vector. But, when we say stress, then you
have to be very clear. If I mean state of stress at a point then it is a second order
tensor. Stress can also be expressed as a vector. There is a relationship between the
two; we will come to that in a minute. Is it clear so far? Now what is this relationship? Why have I
defined this state of stress at a point? In order to understand that let us do a simple
derivation and see whether we can utilize this definition in a much better fashion.
We will take back or let me take you to the same point P. Now, let me see how to determine the stress
vector along a normal n at this point P. In order to do that, let me construct a small
tetrahedron with the height h, with height h about this point P. How am I going to construct
a tetrahedron, small tetrahedron at this point P. Now, let me shift the origin or rather let
me draw the coordinate system say at that point P. That is the point P and let me the
draw the plane of interest whose normal is n at a distance h. So, this perpendicular
distance, the perpendicular distance between this point P and this plane is called as h,
the height of the tetrahedron. The tetrahedron is now made up of these are the axises. What
are they? They are say x1 x2 and x3. This tetrahedron is made up of planes which are
normal to x1 x2 and x3 as well as the plane of interest whose normal say is n, whose normal
is equal to n. Now, let me say that the area of this plane which I have taken to construct
the tetrahedron be say delta S; the same concept as before, only thing is I have just shifted
it a bit. Yes, I know you have an objection, so, let
me say, let me look at the result as h tends towards zero. What is the result as h tends
to zero? When h tends to zero, this plane whose normal is n will pass through the point
P. Let me say that the three planes whose normals are in the x1 x2 and x3 direction
has an area equal to delta S1, delta S2 and delta S3 respectively or in other words delta
S1 is the area of the plane that is formed, that plane whose normal is along x1 and delta
S2 is the plane whose normal is in the two direction and delta S3 is the plane whose
normal is in the, which direction? Third direction. The relationship between delta S1 and delta
S, delta S2 and delta S and so on is very clear. Please note that you can define the
area also as a vector and call this as delta S is equal to n delta S, so that delta S1
can be defined as delta S1 can be defined as n1 delta S. Resolving this in the one direction,
I will get delta S1 is equal to n1 delta S. Similarly delta S2 is equal to n2 delta S.
What is n? Please note that it is normal which results in the direction cosines n1 n2 and
n3. I am sure, yes, all of your familiar with this; so, I am not going to repeat what it
is. So, delta S1 is equal n1 delta S, delta S2 is equal to n2 delta S and delta S3 is
equal to n3 delta S. Now, please note these three planes, the n1, n2 and n3 planes are
also planes which I have used to cut the piece. So, I have taken this piece out. Actually,
what is that I have done? I have taken the piece out from the original body. Now, in
these planes also forces will be acting, the body is to be in equilibrium, so, forces will
be acting in these three planes as well. Let the force that acts on these three planes
be denoted by T1 T say Te1 Te2 and Te3. Let us now look at the equilibrium of this body;
let us look at the equilibrium of this body. How do I now look at the equilibrium of this
body? What do I mean by equilibrium of this body? What is that I say for equilibrium of
the body or what is the condition that I should apply? Yes, correct. So F sigma of the force
vector should be equal to zero. What are the forces that are acting on this body? What
is the body that is of interest to me? It is this body, it is this body that is of interest
to me. This body consists of three planes and an
inclined plane. The forces that are acting in the three planes are Te1 Te2 Te3. They
are vectors. I need not do it every time plus a force that is acting on this plane which
is say Tn. Apart from this there can be one other force. What you thing is the other force
that can act apart from these three? Yes, they are what we called as body forces. The
gravity force is one type of body force, correct. So, apart from these forces the other force
that can act is body force, which typically is a gravitational force. Now, what is gravitational
force? How is it defined? Let me say that gravitational force is defined
by say vector; let me call this vector as P, vector P and defined as body force vector
per unit volume. In other words the force vectors, these are all vectors. So, the body
force vector is defined as P into volume of the body which can be said or which can be
determined as delta S into h divided by 3; one third delta h into S or is it delta S
into h? gives me the volume of the body. This is the body force vector. That is the body
force vector. So, sigma of F is equal to zero. What does it mean? It means that this T plus
this T that is Te1 plus Te2 plus Te3 plus Tn plus that is equal to zero. All the sigma
of F or sigma of all the forces are equal to zero. That is what is meant by the equilibrium
of forces. Just wait a minute, we will write that and before that let us make a small approximation
so that we need not write it again and again. What is that approximation? It is not an approximation,
actually we remove that h tends to zero. What is that force which will be affected
by h tending to zero? Body forces; so, this will go to zero when I put h is equal to zero.
Let me express then Tn in terms of other three forces. The directions are taken care of;
let us not worry about that. We know the directions are taken care of are T of n into what? This
is stress, this is the stress vector; that multiplied by area, delta S is equal to Te1
into what is delta S1? What is delta S1? n1 delta S; correct. So, into n1 delta S plus,
what is that next term? Te2 into n2 delta S plus Te3 into n3 into delta S. This equation
is the equation which depicts the equilibrium of this small tetrahedron which I have taken
to bring in my stress tensor. Immediately the trick is out of the bag. What is that
you are going to do now? What will you do now here? Yes; that is it. So, you resolve this in x1
x2 and x3 direction. So, resolving this in the three directions say for example in the
first direction what is that I will get say T1, T1 n and canceling delta S throughout
on left hand side and the right hand side what is that I will get? T1 n is equal to
T1 e1 into n1 plus T2 e2 into n2 sorry e1 into n2 plus T3 e1 into, no, no, e2 e2; both
of them are T1; T1 e2 and T1 e3. Yes, that is good; so, just as we did in the previous
one, same, there is no difference between the two. The same way you can write T2 and
T3. Before we go to the next one let us rub this
off. We will have the result written in a slightly different notation. What is the notation
we introduced for this? Sigma. What is this? What is this? Sigma11. What is this? Sigma21.
What is this? 31, correct. Hence I can write T1 of n as sigma11 n1 plus
what is the second term? Sigma21 n2 plus sigma31 n3. Same way I can write T2 and T3. How will
you write that? What will happen to this? n2 and so on; sigma21 n2 and so on. I am not
quite lazy to write all this. So, the first question you will ask is how do I simplify
this? I simplify this using what is called as an indicial notation, by using indicial
notation. In indicial notation I will write, I will explain that may be in next class;
let us write this first. In indicial notation, I will write Ti is equal to sigmaji nj. In
your earlier classes you would have studied about summation convention. So, you can say
that summation from j is equal to 1 to 3. For example when i is equal to 1, I get using
the summation convention, what is that I get? I get the equation which I have written here. For example from what you know already you
would have written this equation as T1 and 1 here. The next equation when I want to write,
you would just replace this 1 by 2 and so on. In other words by your summation convention,
you would have written three equations. Einstein introduced a simple notation called indicial
notation and he introduced a convention using the indicial notation. This convention is
called as Einstein convention. I am not sure whether really Einstein introduced it, but
the convention goes by Einstein’s name. Under that convention we do not write down
the summation, but we put certain rules for summation. As you see it, you immediately
see one thing. There is a term here with repeated index. Index is repeated; j is repeated in
that particular group or assembly and i appears on the left on side and the right hand side.
Some index are repeated and some index are free. We will learn more about these two indices
in the next class. What these indices mean and how we use the summation convention, we
will see that in the next class.
