Understanding the strain energy function invariant term

[fruitkiwi]

Hi, Dear all,

Facing problem to understand strain energy function invariant terms
A typical strain energy function consist of strain invariant can be defined as followed
W(I1,I4)=C0+C1(I1-3)(I4-1)+C2(I1-3)^2+C3(I1-4)^2+C4(I1-3)+C5(I4-1),
I1 and I4 are so called invariants of Green's strain tensor. (large deformation)
I1=trC=λ1^2+λ2^2+λ3^2
I4=N1*λ1^2*N1+N2*λ2^2*N2+N3*λ3^2*N3.

here is the complete link taken http://www.engin.umich.edu/class/bme456/ch6fitelasticmodelconstant/bme456fitmodel.htm

1. I read from article that N is a unit vector along the stretch direction, so can i conclude that
I4 consist of unit vector multiply with principal stretch?

2. the lamda in the formula is stretch ratio or principal stretch?

 

[jfy4]

After giving it a quick read,

2)$\lambda_1 ,\lambda_2 ,\lambda_3$ are the "normal stretch", where the link calls them that. but I think they are just the stretch coefficients for force in the same direction as the normal plane vector.

1) Yes, It seems so then, assuming you have "principle"$\iff$"normal"

That's my take on it.

 

[fruitkiwi]

After giving it a quick read,

2)$\lambda_1 ,\lambda_2 ,\lambda_3$ are the "normal stretch", where the link calls them that. but I think they are just the stretch coefficients for force in the same direction as the normal plane vector.

1) Yes, It seems so then, assuming you have "principle"$\iff$"normal"

That's my take on it.

Hi, jfy4,

I actually try to read more, but cannot find resources.
1. so all the N1, N2, N3,should always equal to 1? or under any condition they will change?
2. or can I call them as right stretch tensor?sorry, as i cannot differentiate left and right stretch tensor, so cannot evaluate more for you.

 

[jfy4]

well, $N_i$ is a unit vector, a vector whose magnitude is 1. That is $N_i N_i=N_{1}^{2}+N_{2}^{2}+N_{3}^{2}=1$. But, it is a vector, not a constant, so it's direction can vary. It says:

where $N_i$ are the components of the unit vector aligned along the local muscle fiber direction.