Why need 4th order stiffness tensor expression?

[Galbi]

OK. First of all, I'm novice at Physics so this question may be weird.

Above, there are 2 expressions for strain-stress relations.

Let's assume that all components in the matrix are variables, not zero, not E, nor not G in the first picture.

The first one is written in 2D matrix form, whereas the other one is written in 4th order tensor form although this is a Voigt notation.

Here are my questions:

1. Are these 2 expression same, I mean if the components of each matrix are variables, not zeroes as seen in the picture? That is, can the first form present a general material, like anisotropic?

2. If so, then why do we need to use 4th order tensor expression? The first one is easier to understand. and more intuitive. Why do we learn about the second form?

3. I learned if the material is anisotropic there are 21 independents in the 4th order stiffness tensor. What are the 21 independent variables in the first form? Can you draw circle in the picture or indicate by using notation? I already circled 9 independent components. Where are other 12 components? And why are they independent?

4. Oh, I see the first and second expressions are almost same except shear parts. Can I say that the first one is also Voigt notation?

Thank you very much for reading my question.

 

[Orodruin]

1. No, they are not the same. The first expression has made additional assumptions on the symmetry properties of the material.

2. Answered by 1.

3. You did not circle 9 independent components. There are relations among those elements.

4. Yes, they are both in Voigt notation, but the first one makes additional assumptions about the material.

 

[Galbi]

Thank you very much!