# Is a stress tensor still symmetric when the object is rotating?

I am trying to simulate a spinning & flying deformable football with FEM method. It is always accelerating, instead of keeping static.

Let an undeformed nodal position on this football $$X \in \Omega^0$$. Its deformed position being $$x \in \Omega^t$$ after time $$t$$.

# strain tensor

Now I use the Green-Lagrange strain $$\epsilon$$ to model strain status at ANY point: $$\epsilon_{ij} = \frac{\partial x}{\partial X_i} \cdot \frac{\partial x}{\partial X_j} - \delta_{ij}$$

It is absolutely symmetric.

# stress tensor

And if the constitutional model is linear, we can easily build the stress $$\sigma_{ij} = C_{ijkl} \epsilon_{kl}$$

Is $$\sigma$$ still symmetric? Note that now any infinitismal area is accelerating, no moment equaibrilum can be satisifed.

# My questions

1. Is stress tensor $$\sigma$$ still symmetric? If so, why?

2. If not, why everybody claims it is symmetric? Or will it break some physics law?

3. Is there any useful constitutive model (elastic potential)?

• I am not sure if you mean this transformation, in which case you find that $\sigma'^T=(A\sigma A^T)^T=A\sigma^T A^T=A\sigma A^T = \sigma'$. Commented Jun 5, 2022 at 12:15
• The Cauchy stress located in the infinitesimal stress analysis, in another word, when the system is under very small deformation. Maybe we can only use it when "The Cauchy stress tensor is used for stress analysis of material bodies experiencing small deformations:", as the wiki said. The football in my scene is under big deformation, where finite strain theory must be applied, instead of this infinitesimal strain theory. Commented Jun 5, 2022 at 12:34

## 1 Answer

I will answer your questions below.

1. Yes. The Cauchy stress is still symmetric. The thing is that this stress measure is not energy conjugate with the strain measure that you picked. This means that the double contraction between your stress and the strain (rate) tensor does not represent the work (rate) done to achieve that deformation level.

2. It is still symmetric but you should pick another pair of strain-stress. The "right" stress tensor for your strain measure is the second Piola-Kirchoff stress tensor. I think that the Cauchy stress tensor does not have a conjugate pair, though (see 1).

3. That depends on the material that you are using. My first guess is that the football can be modeled with a hyperelastic material (see 2).

### References

1. Hoger, A. (1987). The stress conjugate to logarithmic strain. International Journal of Solids and Structures, 23(12), 1645-1656.

2. Karimi, A., Razaghi, R., Navidbakhsh, M., Sera, T., & Kudo, S. (2016). Measurement of the mechanical properties of soccer balls using digital image correlation method. Sport Sciences for Health, 12(1), 69-76.

• Thanks sir. I am learning to understand your answer, which may take 1~2 days. I will response as soon as possible. Commented Jun 9, 2022 at 3:12