# Negative Poisson's Ratio in Rotated Orthotropic Material

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1. Oct 23, 2014

### jester117

Hi all,

I'm looking for physical and mathematical explanations for a phenomenon I've noticed when working with rotated material properties relative to an XYZ coordinate system. For certain rotations (~40 to ~50 degrees) about a single axis the S12, S13, or S23 components of the compliance matrix become positive, which implies that the corresponding Poisson's ratios are negative. The material I'm trying to simulate is a single crystal metal, so I wouldn't expect a negative Poisson's ratio.

I'm able to simulate this result across multiple methods for basis transformation, so I doubt there's an error there. I've also been able to confirm that the stiffness-compliance matrix inversion is correct.

If you'd like to simulate the problem, you can use this site: <http://www.efunda.com/formulae/solid_mechanics/composites/calc_ufrp_cs_arbitrary.cfm>. I used Msi units, both Young's moduli = 18, shear modulus = 18, Poisson's ratio = 0.38, and theta = 45.

Of relevance:
The form of the compliance matrix used, http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/hooke_orthotropic.cfm
Relevant articles,
http://arc.aiaa.org/doi/abs/10.2514/3.4974
http://appliedmechanics.asmedigitalcollection.asme.org/article.aspx?articleid=1415488

Thanks for looking into this,

jester117

2. Oct 28, 2014

### Greg Bernhardt

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Oct 29, 2014

### jester117

Thank you for the nudge, haha. I do have additional information.

The problem I was having was concerned with the rotation of a 6x6 stiffness matrix for 3D stress-strain vectors. The coordinate transformation through Euler angles and the corresponding DCM generated were both correct (i.e., rotating the 3x3 coordinate system matrix was correct). However, the problem was with rotating the 6x6 matrix into the new coordinate system. I changed the rotation function to follow the structure given here in chapter 6.7 of the following link:

http://www.ae.iikgp.ernet.in/ebooks/ [Broken]

The problem with the negative Poisson's ration disappeared, although a new one popped up in it's place. For this problem, I think I'll post a new thread since it's only tangentially related. I'll post another reply with the link to the new thread for anyone interested.

I still can't explain why the efunda plane stress site I linked in my original post leads to a negative Poisson's ratio.

Thanks,

jester117

Last edited by a moderator: May 7, 2017
4. Oct 29, 2014