Negative Poisson's Ratio in Rotated Orthotropic Material

[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

 

[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?

 

[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/

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

 

[jester117]

Link to new thread regarding anisotropic elasticity properties.
https://www.physicsforums.com/threa...-properties-for-anisotropic-materials.778893/

 

[alphy]

Hi jester117,

Thank you for bringing this interesting phenomenon to our attention. I can provide some insights and explanations for the negative Poisson's ratio observed in rotated orthotropic materials.

First, let's define what a Poisson's ratio is. It is a measure of the transverse strain (change in width) of a material when it is stretched or compressed in the longitudinal direction (change in length). In a typical material, the Poisson's ratio is positive, meaning that the material will become thinner when stretched and thicker when compressed. However, in some materials, the Poisson's ratio can be negative, meaning that the material will become thicker when stretched and thinner when compressed.

Now, let's consider the case of a rotated orthotropic material. Orthotropic materials have different properties in different directions, and when rotated, these properties change accordingly. In your case, you have observed that for certain rotations, the compliance matrix (which describes the relationship between stress and strain) results in positive values for S12, S13, or S23 components. This indicates that the material becomes thicker when stretched in certain directions, leading to a negative Poisson's ratio.

This phenomenon has been observed and studied in various materials, including single crystal metals. It is believed to be a result of the microstructure and arrangement of atoms in the material, which can lead to unusual mechanical properties. In your simulation, the Young's moduli and shear modulus were kept constant, but it is possible that these values may also change with rotation, contributing to the negative Poisson's ratio.

In terms of mathematical explanations, the compliance matrix used in your simulation is based on the Hooke's law for orthotropic materials, which assumes that the material is linearly elastic. However, it has been shown that in certain cases, non-linear behavior can lead to negative Poisson's ratios in rotated materials. This could be one possible explanation for the phenomenon you have observed.

I would recommend further research and experimentation to fully understand and explain this phenomenon. The articles you have provided are a good starting point, and I encourage you to continue exploring this topic. Thank you for sharing your findings and for stimulating scientific discussion.