It will be presented by the University of Erlangen Nürnberg.

Okay, so good morning again.

So we'll again start with where we left.

We were working on constitutive equations for the rod.

And the idea was we wanted to relate the forces and moments with the kinematic variables,

isn't it? Because the equations that we had derived n prime plus n hat equals 0 and m prime

plus r prime cross n plus m hat equal to 0. So, these equations have n and m and we don't know

how they are related to small r and capital R, the kinematic variables. So, that's why we are

talking about constitutive equations. And then we use frame indifference and finally came to

this form of constitutive equation psi of r transpose small r prime comma big R transpose

big R prime. So, that is the strain energy or stored energy per unit undeformed length.

Right and then we also said what is the physical meaning of this? So, if we say this is V, then

these were actually the components of little r prime which is the centerline tangent in the

frame of directors and we also found that this if we write it as V1, V2, V3, then the first two

components are the shears and the last is axial stretch. And then we wanted to understand the

meaning of the other argument over here. So, that is the first argument. Now,

we want to understand the meaning of the second argument. Now, as rotation, if this is a rotation

matrix, this quantity is going to be a skew symmetric matrix, isn't it? Something that we

had worked out last time. I can also show you very quickly. So, r transpose r is equal to identity.

So, then you take the derivative on both the sides and you get r transpose prime times r plus r

transpose r prime equal to 0 which means r transpose r prime is equal to negative of r transpose r prime

transpose. So, here you got a matrix which is negative of its transpose, it is a skew symmetric

matrix. So, that means it is a skew symmetric. So, let us define it as big K, which is r transpose

r prime and we write it as 0 0 0 minus K 3 K 3 K 2 minus K 2 minus K 1 and K 1.

And the way reason I define it like this is because its axial vector. So, the axial vector

of K which we denote as little k will then come out to be simply K 1 K 2 K 3. So, that is my axial

vector. So, that is the reason why I define my big K like this and what is the relation between

this big K and its axial little k is that if you say big K times any vector a then it is simply

its axial vector K cross a. So, now we want to understand the meaning of these three quantities

here what is K 1 what is K 2 what is K 3 and we will see that they in fact they in fact are

bending curvatures and twist K 3 for example, is twist and K 1 K 2 are bending curvatures.

Let me give you the example of twisting. So, if I say twisting of a rod. So,

we if we have a circular rod for example, that is a circular rod and if you twist it

no then the axial lines become helical is not it. So, it is going to become something like this

and what will be the rotation of any cross section if I think of a cross section over here it is

simply going to rotate about this axis which is e 3 and if I say theta is that rotation then

that is simply going to be some number let us call that let me call it K 3 only let me call it K 3

so it will some number K 3 times s the arc length right biggest and then we know that this K 3 is

nothing but the twist right from our strength of materials course. So, K 3 is actually the twist.

So, what will be my rotation matrix big R what is that going to be for this particular case. So,

since the cross sections are all rotating about e 3 axis the rotation matrix simply becomes 0 0 1

0 0 cosine of K 3 times s minus sine of K 3 times s then sine of K 3 times s and cosine of K 3 times

s right is not there is a rotation matrix and then if you talk about your little r what is your little

r if there is no stretching if you are just twisting then little r is simply s times e 3 is not it is

simply s times whatever it was earlier because I am not stretching it maybe I should write capital

S here then what is then little r prime it is simply e 3 and what is big R prime when you do big

R prime these ones all become 0 right cosine of K 3 s that becomes sine of K 3 s times K 3 right

minus K 3 yes. So, this becomes minus with a minus sign minus sine of K 3 s and you get a K 3 in the

front because each of them is going to cough out K 3. So, K 3 comes out and minus sign of K 3 s

becomes simply minus cosine of K 3 s then sine of K 3 s becomes cosine of K 3 s and cosine of K 3