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.

 

[PJC01]

3. The strain energy function is a mathematical representation of the energy stored in a material due to deformation. In this case, the strain energy function is defined by the invariants of Green's strain tensor, which are mathematical quantities that describe the deformation of a material.

4. The first invariant, I1, is the trace of the Green's strain tensor and represents the total amount of deformation in a material. It is calculated by taking the sum of the squares of the principal stretches (λ1, λ2, λ3).

5. The second invariant, I4, is a combination of the principal stretches (λ1, λ2, λ3) and a unit vector (N1, N2, N3) that represents the direction of the stretch. It is used to account for the anisotropic properties of a material, where the material's response to deformation is dependent on the direction of the applied stress.

6. The values of the coefficients (C0, C1, C2, C3, C4, C5) in the strain energy function are determined experimentally and can vary depending on the material being studied.

7. To answer your questions, yes, I4 does consist of the unit vector multiplied by the principal stretches. And the lambda in the formula represents the principal stretches, not the stretch ratio.

Overall, understanding the strain energy function and its invariant terms is important in predicting the behavior of materials under different loading conditions. I recommend further reading and studying on this topic to gain a better understanding.