Okay, so let us continue where we left last time. So we were doing kinematics of a special

cos rat rod. So I will just draw the picture. So here we have our rod in the reference state.

So you see we are going to call the beam from now on we will call it as a rod.

So if I travel a distance s along e3 axis I find a cross section over here, a planar

cross section and that would have gotten mapped somewhere over here with its centroid at r

of s, small r of s and then we also had the two directors in the plane of the cross section

d1 of s and d2 of s. So the kinematic variables are small r of s and the two directors and

then we said that if we are talking about a general cos rat rod in that the directors

need not be unit normed or orthogonal. So and that was allowing the cross section to

stretch in certain directions or allowing what is called cross sectional shearing. So

if I have to just draw that again here is a cross section that is your e1 and e2 and

that gets mapped to some electrical cross section with d1 and d2 and then it not be

perpendicular. So I will just redraw one of them to emphasize that they are not perpendicular.

And then we also thought about a third one d3 but this is not an independent variable

d3 of s is chosen in such a way that it is perpendicular to d1 and d2. So it is not independent,

not independent but it turns out to be a convenient vector because it is perpendicular to the

cross section and defines the cross section normal. So it is not independent but defines

cross section normal. And we also said that so this was a general cos rat rod but if we

confine ourselves to what we want to learn that is a special cos rat rod then the directors

are unit normed and also orthogonal to each other. So what that means is that now your

d1, d2, d3. So for a special cos rat rod D1, D2 and D3 are now formed and orthonormal

normal triad. So, we have first in the reference state e 1, e 2 and e 3 which gets mapped to d 1,

d 2, d 3. So, we have e 1, e 2, e 3 one orthonormal triad getting mapped to d 1, d 2, d 3 and since

they are two orthonormal triads it can always be achieved using a rotation tensor. So, what that

means is that your d i of s is then simply that rotation tensor R of s times e i. So,

this is also you know sometimes called director basis or local basis because they are attached

to different cross sections and each cross section has its own local basis d 1, d 2,

d 3 everywhere whereas we have another basis e 1, e 2, e 3 which is a global basis that is right

here it is the same for everyone. So, and then the constraint map that we had was f of x 1,

x 2, x 3 that is equal to and x 3 was equal to s small r of s plus x alpha d alpha,

but now that becomes. So, the same thing now becomes plus r times x alpha e alpha right

because d alpha is r times e alpha and alpha here always goes from 1 to 2 whereas i here is

going from 1 to 3. So, that is my constraint map now for the spatial cross-strand rod and you see

what this is telling us here we have the first part which takes any cross section. So, we have

a cross section at s it takes that cross section and just translate translates it by that r of

little r of s once the cross section has translated then you rotate it by big R right.

Then you see what you have within here it is just the cross sectional coordinates. So,

this part does the job of rotating a cross section whereas this part does the job of

translating the cross section. So, every cross sections are just getting translated and rotated

and of course if you just look at this map then the cross sections are rigid cross sections are

not deforming. So, that is why it is called a constraint map. So, this translates whereas

this part rotates. So, once we have our constraint map then you can use your 3D

elasticity and substitute this map in the 3D elasticity equation and integrate it over the

cross section to obtain some differential equation for your unknown small r and big R.

So, you see now the let me just rewrite the unknowns are small r and big R. So,

these become your kinematic variables for the special cos root rod. So, that is so our goal

would be to get equations solving which we can find small r and big R.

And let me make a slight departure here from this kinematics you see this kinematics is not so good

because it keeps your cross section rigid is not it. Just think of a rod a very simple guess I have

got a rod and I am just stretching it does the cross section remain rigid it does not right it