Which strain tensor is used in Föppl–von Kármán equations?

[MarkoA]

Hi,

I have a question concerning the von Kármán equations. I want to better understand the compatibility relation. The wikipedia article states that:

https://en.wikipedia.org/wiki/Föppl–von_Kármán_equations
"The components of the three-dimensional Lagrangian Green strain tensor are defined as .."

What follows looks much like the Green-Lagrange strain tensor. But, as I understand the Green-Lagrange strain tensor is used in the total Lagrange formulation and thus not compatible to the Cauchy stress (which is why one would use the 2nd Piola-Kirchhoff stress tensor).

Is the Green strain tensor as written in Wikipedia different from the Green-Lagrange strain tensor? The wikipedia-Link to finite strain theory didn't help.

Thanks,
MarkoA

 

[Achy47]

Dear MarkoA,

Thank you for your question regarding the compatibility relation in the von Kármán equations. The Green strain tensor mentioned in the Wikipedia article is indeed different from the Green-Lagrange strain tensor. While both tensors are used in the study of finite strain theory, they have different applications and interpretations.

The Green strain tensor, also known as the Lagrangian strain tensor, is defined as the symmetric part of the displacement gradient tensor. It is used to measure the local deformation of a material in a small region, and is commonly used in the Eulerian description of fluid mechanics. This tensor is compatible with the Cauchy stress tensor, meaning that they can be used together in the same formulation without causing any inconsistencies.

On the other hand, the Green-Lagrange strain tensor is defined as the symmetric part of the deformation gradient tensor. It is used in the total Lagrangian formulation, which takes into account the deformation of the entire body. This tensor is not compatible with the Cauchy stress tensor, and that is why the 2nd Piola-Kirchhoff stress tensor is used instead.

In summary, the Green strain tensor and the Green-Lagrange strain tensor are two different tensors with different applications. The Green strain tensor is used in the Eulerian description and is compatible with the Cauchy stress tensor, while the Green-Lagrange strain tensor is used in the total Lagrangian formulation and is not compatible with the Cauchy stress tensor.

I hope this helps clarify your understanding of the compatibility relation in the von Kármán equations. If you have any further questions, please do not hesitate to ask.