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.