# Using 2nd-Piola-Kirchoff stress rather than Cauchy

• hoomanya
In summary, the conversation discusses the use of the 2nd Piola Kirchoff Stress, S, instead of the usual Cauchy stress, sigma, in a general transport equation of a material. The speaker clarifies that S and the Lagrangian strain tensor, E, are work-conjugate pairs due to their invariance to rigid body rotation. However, S is only meaningful for small strains and should not be used with E in a constitutive relationship. The speaker also mentions the importance of considering the constitutive relationship in rate-form, and acknowledges that their perspective is from an FEA standpoint.
hoomanya
Hi,
I need to use the 2nd Piola Kirchoff Stress (S) rather than the usual Cauchy stress i(sigma) in a general transport equation of a material. I was wondering if I need to replace the stress tensor (usually denoted by sigma) with S, or do I just write sigma in terms of S using the relationship that exists between S and sigma.
Thanks!

I see you have no responses, so I'll give you something to think about -- though, I speak from an FEA standpoint:

Why do you NEED to use the PKII stress, S?
Is it because you are using the Lagrangian Strain tensor, E?

Your stress and strain pair (constitutive relationship - e.x. Hooke's Law) must be work- conjugate, meaning that they must behave the same under rigid body rotation. The PKII stress tensor, S, and the Lagrangian strain tensor, E, are indeed both invariant to rigid body rotation, so they are a work-conjugate pair.

S is only meaningful for small (infinitesimal) strains. You could potentially acquire the Cauchy stress, $\sigma$, from S, as you describe -- but you wouldn't want to use $\sigma$ with E in your constitutive relationship. Keep in mind that $\sigma$ is defined in in a spatial reference frame (it is defined in your original coordinate system, regardless of whether there is rigid body rotation).

If your constitutive relationship is in rate-form, then that's a whole 'nother story.

Also, I remind you that I am an FEA guy, so if nothing I said applies to you, I apologize.

Thanks. What you said made sense. I also had a read of the wiki page on stress and am a lot more clear now.

## 1. What is 2nd-Piola-Kirchoff stress and how is it different from Cauchy stress?

The 2nd-Piola-Kirchoff stress is a measure of stress in a material that takes into account the material's deformation. It differs from Cauchy stress, which only considers the external forces acting on the material. 2nd-Piola-Kirchoff stress is considered a more accurate measure of stress in nonlinear materials.

## 2. When should 2nd-Piola-Kirchoff stress be used instead of Cauchy stress?

2nd-Piola-Kirchoff stress should be used when dealing with nonlinear materials, as it takes into account the material's deformation and provides a more accurate measure of stress. It is also commonly used in finite element analysis and computational mechanics.

## 3. Is 2nd-Piola-Kirchoff stress always larger than Cauchy stress?

No, 2nd-Piola-Kirchoff stress can be larger or smaller than Cauchy stress depending on the material's deformation. In some cases, they may be equal.

## 4. How is 2nd-Piola-Kirchoff stress calculated?

2nd-Piola-Kirchoff stress is calculated using the material's deformation gradient and its Cauchy stress. It is derived from the principle of virtual work and is represented by a second-order tensor.

## 5. Can 2nd-Piola-Kirchoff stress be used in all materials?

No, 2nd-Piola-Kirchoff stress is primarily used in nonlinear materials. In linear elastic materials, Cauchy stress is typically used as it provides an accurate measure of stress. However, in some cases, 2nd-Piola-Kirchoff stress can still be used in linear materials to account for small deformations.

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