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Okay, so good morning everyone.

Let's do a recap of what we have done

about the special cross rat rods. Very quick recap.

So first the kinematics.

So we thought of a center line.

So that was the center line of the rod.

And we associated two variables with every point of the center line. That's r of s and r of s.

So r of s was to denote the center line of the deformed rod.

And r was to denote the average orientation of the cross section.

So that's the kinematics.

We also said that we extended this to the whole rod by saying

x of x1, x2, s is equal to r of s plus r of x alpha e alpha.

So this was the constrained kinematics. So that's the constrained kinematics.

You can say constrained kinematic extension to the whole rod.

But we didn't really use this later on in our theoretical development.

So you know we haven't used that. So I would like to emphasize that we haven't used that in our subsequent development.

So again we started with r, capital R kinematics.

Then we obtained balance laws.

Remember you took an arbitrary segment of a rod and you did force balance and moment balance.

And you obtained the two equations n prime plus n hat equal to 0.

Then m prime plus r prime cross n plus m hat is equal to 0.

And while deriving these balance laws, you didn't use this kinematics.

We never used this kinematics while deriving the balance laws.

Once we had the balance laws, then we saw that we have more unknowns than equations.

So that's where we started thinking about the constitutive model.

Then we thought of constitutive law.

And we postulated a strain energy per unit undeformed length of function.

And we said that it must depend on the kinematics of our theory and its first derivative.

Remember we started with this.

And then we used frame indifference, material frame indifference.

Or you could also say objectivity to come up with this function.

And then we said what are the physical meanings of these two quantities, v and k.

V contains shear and axial stretch and k contains bending curvature and twist.

So v for example was big R transpose little r prime and k was axial of big R transpose big R prime.

And then maybe I could write here itself.

We then related the derivatives of this strain energy with forces and moments.

Isn't it? And that again was done without resorting to this kinematics.

Isn't it?

We wrote down the full energy of the rod to the first variation, got the Euler Lagrange equation.

And compared that with this balance equation.

And found out that your little n is equal to big R times del psi over del v.

And little m is equal to big R times del psi over del k.

And we also saw that del psi over del v have shear forces and axial force.

And del psi over del k have bending moments and twist.

After this we of course wrote down our balance equations, the 13 L system of 13 first order ODE.

Isn't it?

And then we talked about material symmetry in elastic rods.

And we said, I mean we derived that if the rod has got the SO2 symmetry, SO2,

then it has the following invariants and the energy must depend on those invariants.