Time Frame Objectivity
So, welcome back everybody to another lecture Introduction to Soft Matter. As we are
drawing to a close, we were discussing the idea of Objectivity. Why we are discussing that it
will be hopefully clear by the end of this class, okay.
(Refer Slide Time: 00:50)
So, we were discussing this idea of what is called frame invariance objectivity. And the basic
idea is that the fundamental laws of nature should be independent of frame of reference. So,
the basic idea is that the fundamental laws of nature, here we imply laws of physics. Laws of
nature are independent of the frame of reference.
And we said that another 2 frames are equivalent if they are related to each other. They
understand how to calculate, 2 frames are equivalent if they have an agreement on the ideas
of length, time and the sense of time.
And 2 different frames, which are related to each other by a rigid body motion can be given
by this kind of a transformation, where this is the these are * x represents the transformed
variable or the transformed components, let us say of a position and c(t) is a translation
given by a rotation and here you have the older position or the values that we found in the
older system * x = Q(t)(x − o) + c(t) where Q(t) rotation and c(t) represents translation.
(Refer Slide Time: 03:11)
So, we came up with idea of objectivity then, and I will just review a very nice statement. This is again following up with what we discussed last time. These are again from the notes of Professor Suo and he frames it very nicely where he says that here the fundamental hypothesis, here is the final fundamental hypothesis. The rheological behavior of materials is independent of rigid body motion.
Independence of rigid body motion is equivalent to the independence of the choice of frame. You can choose any frame of any kind as long as it is rigid. And then he inserts this funny anecdote where supposedly Henry Ford is famous to have said that you can choose a car of any color as long as it is black, right.
So, you can choose a frame of reference as long as it is rigid. And this idea that the fundamental this behavior is independent of rigid body motion is a hypothesis. So, whether this rheological behavior of a material is truly independent or not, is ultimately settled by experience. We do not know the rheological behavior of, we do know that the rheological behavior of most materials is at least insensitive to rigid body motion.
And here we are all discussing velocities etc., much smaller than the speed of light. So, we are not discussing any relativistic, we are not taking into consideration relativity here.
(Refer Slide Time: 04:45)
And, I also discussed that this idea of ()Qt which is a transformation which gives you rigid
body motion where Q is a rotation. And then we also used the idea that T QQ I = . Now this
is from the book, the mechanics and thermodynamics of Continua Gurtin, written by 3
authors Gurtin, Fried and Anand.
And they say that there is some disagreement as to whether only rotations or all orthogonal
tensors should be employed in the statement of frame indifference. For example, Trusdell and
Noll, 1965 stayed the principal with ()Qt an orthogonal tensor. While Chadwick, 1976 and
Gurtin require only that Q(t) be a rotation.
And then they finally go on to clarify that, according to another work, we believe that Q(t)
should only be a rotation. And here it is important to just point out that you can have an
orthogonal transformation which is not a rotation, so you can have a reflection. So, you can
have Q(t) equal to minus 1, the determinant of Q(t) and that would reflect that would be a
reflection. And that is not to be taken into account in this sense.
Now, this is only an introductory course, so we are only brushing up or lightly touching upon
some very-very complicated topics. And I would like to point out that these topics are
basically something that are in the realm of advanced tensor mathematics and advanced
continuum mechanics. So, we are not dealing with, we are not going through the proofs,
very-very rigorous proofs of everything.
We are just lightly taking up some of the simpler ideas that are inherent in this and you are more than welcome. Here the idea there is a specific reason why we are discussing objectivity and you will figure that out by the end of this class.
But if you want to understand this in more detail you would have to take a different lecture or a series of lectures which goes through as mathematically very rigorous. Okay so, let us return back to our notes.
(Refer Slide Time: 07:20)
So, before we discuss further let us understand a few terms. So, in mechanics we assume that
there is a body and this is something called the reference space. The reference space can be
arbitrarily set but it is basically a situation from where you start comparing all your
deformations that are going to occur later.
So, this body B has some material point in it, let us say X . And this later on is going to go
through a series of deformations. So, this body B is now going to go through a series of
deformations in time. So, that is why I put this index t B , so that this is at a later point.
And then this particular point here is going to map to some other point now. And let us call
that x . And let us say that is equal to some function x = m(X, t) . Okay so B , this is B is the
reference configure, B is also called the reference configuration. The name itself should alert
you about the purpose of this, this is the reference state with which we are going to calculate
others and this capital X , this is also called a material point or a particle.
The x is a spatial point, which is basically a mapping of this X to at another time in the same
body. So, x is also called a spatial point occupied by X . And m is such a function that it
gives you the motion of the body. And at any given time, so, if you freeze the time, then tB at
a given time is the deformed state.
Now, we are doing all this because we want to introduce a very specific tensor which is
called the Cauchy stress tensor. And I want to have a discussion on that and basically the
relationship with the 1 D models that we introduced and more generalized models, so to do
that we are going through a series of steps, okay.
So, now we are going to discuss since we are going to discuss stress tensor, I told you last
time that a tensor is a linear mapping of vectors to vectors. So, let us say a tensor say I am
just taking some variable S, and then I am putting this double bar here is a linear mapping can
be, a tensor can be looked in many different ways, but it can also be considered can be
considered as a linear mapping of vectors to vectors.
So, say u , so let us just say there is a given vector, so u is a given vector. And this S is a
tensor that is giving me this transformation, then when S is going to act u , it is going to
result in another tensor, sorry another vector for me. Let us call that just v . So, by the way,
these are just variables okay where v , S and u , we are just using some names at the moment
to illustrate an example. v S u =
And we will see why this particular form is important. Now Cauchy’s hypothesis, so since we
are going to discuss the Cauchy’s stress tensor, so now we have understood to some extent or
in a very simplistic manner, we have an understanding of what tensors are.
(Refer Slide Time: 12:55)
And Cauchy’s hypothesis concerns the form of what is called contact forces is about and he
introduced, Cauchy introduced, what is called as a traction field. So, Cauchy introduced the
concept of a surface traction field. Let us say that is some t , is actually force that is
associated with a given area, a small area which is an area we know that infinitesimal area is
uniquely determined by its normal vector.
So, let us say n is the normal vector of that, x is the position at which it is being computed
and t is the time, okay. So, I have already introduced a vector traction vector t. So, t is here
also the time and we are but this t is a scalar. So, even though we have the same symbol, we
can still distinguish between the two.
So, this t is basically defined is, so t is defined for each unit normal vector and for each
spatial point in B and for each and for all times, so maybe we just add that and forth. So, in a
sense this traction vector represents the force per unit area exerted on the material.
(Refer Slide Time: 16:02)
So, Cauchy’s theorem which concerns this and Cauchy’s theorem states that there exists a
spatial tensor field and this is actually the stress tensor field and we had already introduced
the symbol sigma before right, so I am just going to say a sigma with a double bar.
So, this is the stress tensor field called Cauchy stress such that, so now you have a tensor
field, right. So, this we know that a tensor has to act on some other vector to give me another
vector. So, this tensor field is now going to act on, you can probably guess, on the normal
vector. And what is it going to give me? It is going to give me this t .
t = n
So, these are spatial vectors because they are in the observed system. And these are again
spatial vectors. Now, assuming that this, these forces are being calculated from the
generalized body forces, you can show that these are going to transform in a way that they are
invariant.
It is probably easy to understand why the normal vector is going to be invariant because the
normal vector doesn’t really depend, going to depend on what reference frame you are using.
It exists irrespective of the particular reference frame you are going to take, because the only
thing it is dependent upon is that infinitesimal area.
And similarly, there is a way to discuss what is called as a generalized body force. And then
you can show that this t bar is also the traction field is also a invariant form. So, they are
going to transform as per the rules that we had already set before.
So, we will just quickly write that down. So, t and n and here in brackets, I will just write
assuming generalized body forces will transform as per the following rules, so we had
already said that this t in a rigid body rotation given a rigid body rotation and translation, the
rule that governs this is going to be and for the normal vector similarly, this is the value in the
transformed coordinates, this is now, the Q is the rotation.
So, you can show that, so from here you can show that the Cauchy stress tensor given the
relationship that we have already found is frame indifferent. So, just let us write that. So, now
your *t is equal to some transformed stress tensor multiplied by the transformed normal
vector. And we already know what these 2 individually transform as, so I am just going to
write that. * t Qt = and
* n Q n =
t* = *n*
* Qt = Qn
So, I am just replacing the previous equations, here. And this implies if you do one
* ( ) T t = Q Q n , this just right. So, in the original coordinate system, so now from this, you
can see that * is and I am skipping one step because all you have to do is to equate the 2
and then multiply with the T Q and what you will get is this, okay.
* T QQ=
(Refer Slide Time: 21:14)
So, this has the implication that the Cauchy stress tensor is frame indifferent or is objective, okay and this is very nice, because why is this nice? PDEs, the ODEs that we have derived. So, for let us say for example, we had this particular equation, right here. Now this was a 1-dimensional system.
So, these are just, we treated them almost as scalars and here this Cauchy’s if you replace this with a Cauchy stress tensor then this forms the state invariant. But you also have a time derivative. So, is the time derivative going to be invariant, right?
We are trying to write an invariant form for the entire thing and on this side you have (the deformation), the strain rate fields. So, will that be frame invariant? Now, we had seen previously last time where we have discuss that the velocity. So, although the separation is an objective variable, but the velocity is not, right.
So, if you just want to quickly review that. So, we had seen last time that separation is framed
indifferent right, that was one of our most important points of discussion and we had said that
the separation between 2 places in the Euclidean space is frame indifferent vector.
The separation is a mother of all frame invariant variables indifferent variables. They have
different fathers, time energy, entropy energy, electric charge, as well as quantity of atoms
molecules and colloids of every species. They are scalars and frame indifferent. But relative
velocity is frame sensitive. So, even though separation is frame indifferent, its rate the
relative velocities frame sensitive. This is what we saw in the last class.
And in general, the rate of a frame indifferent variable is frame sensitive. So, we are just
discussing the reason this last note is important is because we just said the Cauchy’s stress
tensor is objective. But then that rate of the Cauchy’s stress tensor is not going to be
objective, okay.
So, we are going to make a note of that but the rate of change, the time derivative, rate of
change and the rate of change here implies obviously with respect to time. The rate of change
of this is not frame indifferent. And this can actually be proven quite well. So, we will
quickly prove the generalized idea that if, so let us say.
So, let us say that T is an objective tensor field. So, T transforms as the * T T QTQ = , if you
have, if you are going to calculate in a new frame, the components, you know that that is
going to transform as this quantity, where Q is some rotation matrix. So, now, if you take a
derivative of this, so let us say what happens if you take the derivative. So, since this is an
equality, I can take derivatives time derivative of both sides, so I will just indicate by a dot
that we are taking the time derivative.
And here you can start taking time derivatives. And first I am going to take the derivative of
T , I am just going to keep that on the left hand most corner and you will see why. But I also
take that derivative of the others, do by chain rule. So, that becomes Q
•
. Now, if Q is a
rotation, then if Q is a rotation, then this particular tensor is another tensor, which is also
called the spin tensor of the frame.
* T
T T T QTQ QTQ QTQ
• • • •
= + +
So, I can replace these Q
•
’s basically that is what I am trying to do here. So, I can replace this
Q
•
’s. And what I will get in return is
*
T
•
. You can probably already see that it is not that
derivative is not going to be objective because then only this part, you should not have had
this this section, this section is the one that is going to create the problem.
TQQ
•
=
You only wanted this term if it were supposed to be objective, but these terms are going to be
there and you cannot wish them away. And you can show that this is now going to become,
you will have a Q
•
, is no sorry, not Q
•
this is the derivative of t. So, you have this term plus
when you just do one more step, but you will find that this becomes and this extra term is not
going to be, is non-zero, right.
*
T * * T QTQ T T
• •
= + −
So, we have this extra term and this is going to create a problem. So, basically this implies
that
*
T
•
is not it is not objective. So, we just verified the statement, last statement that we had
seen and where we said that in general, the rate of a frame indifferent variable is frame
sensitive. So, if you want to make a constitutive equation such that, so if you want to make a
constitutive equation, which involves a Cauchy stress tensor, the Cauchy stress tensor is not
going to give a problem because it is objective.
But the derivative of the Cauchy stress tensor which we saw happens occurs in the
Maxwellian, the Kelvin Voigt Meyer bodies and its generalizations there is a derivative of
stress is there, the time derivative of stress and that is going to be a small problem because it
is not going to be objective. So, we will see how people resolve this in the next class. Okay,
so we will stop here for today.
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