Poisson ratios for Orthotropic materials (composites)

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Summary:
For an orthotropic material (e.g. carbon reinforced polymer), how to derive the Poisson matrix from 3 values?
I'm new to composite materials. I've studied mechanical engineering but I am actually usually involved in hydrodynamics (in which I've done my masters). However for a project we do fluid structure interaction with composites, and as these things go, you cannot get away with the 'black box' approach (I wish I could sometimes...). So I'm trying to learn the basics.

In the FEM software I use I need to specify three Poisson ratios: ##\nu_{12},\ \nu_{23},\ \nu_{31}## (and also three E and three G moduli). It is orthotropic material (3 symmetry planes). I want to derive the other Poisson ratio's because I want to know if the supplier provided a consistent set of data.

Some questions:
  1. ##\nu_{12}## gives me the strain in 2 direction from the strain in 1 direction, i.e. ##\epsilon_2 = -\nu_{12}\sigma_2/E_2##, correct?
  2. But I believe in general, for orthotropic materials, ##\nu_{12}## is not equal to ##\nu_{21}##. Correct?
  3. If indeed so, can I derive the complete matrix of all Poisson ratios from ##\nu_{12},\ \nu_{23},\ \nu_{31}##? And how?
  4. There are al kinds of constrains that must hold for a real material, e.g. ##\Delta = 1-\nu_{12}\nu_{21}-\nu_{23}\nu_{32}-\nu_{31}\nu_{13}-2\nu_{21}\nu_{32}\nu_{13} > 0##, which is I believe the change in volume...? Are there more of these checks that I can do to find if my input dataset is consistent?
 

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