Poisson's Ratio for Ogden Hyperfoam

How can you determine the Poisson's ratio for Ogden hyperfoam?

The Ogden hyperfoam model is a strain energy function that allows for large compressibility, as the deviatoric and volumetric terms are actually coupled. The "beta" parameter can be related to Poisson's ratio as follows:

For a linear elastic material:

bulk_modulus = EX/3/(1-2*NUXY)
shear_modulus = EX/2/(1+NUXY)

Therefore, the ratio of bulk to shear modulus is:
bulk/shear = 2*(1+NUXY)/(3*(1-2*NUXY))

Per the elements reference for Ogden hyperfoam constants, the initial bulk and shear moduli are:

initial_bulk_modulus = SUM(mu_i * alpha_i (1/3 + beta_i))
initial_shear_modulus = SUM(mu_i * alpha_i)/2

Therefore, the ratio of bulk to shear modulus is:
bulk/shear = SUM(1/3+beta_i)*2

Assuming that beta_i is a constant (i.e., beta_1=beta_2=..., constant for all values of i), we can therefore relate Poisson's ratio with beta_i:

bulk/shear = 2*(1/3 + beta_i) = 2*(1+NUXY)/(3*(1-2*NUXY))
(1/3 + beta_i) * 3 * (1-2*NUXY) = (1+NUXY)
-1+1+3*beta_i = NUXY + 2*NUXY + 6*NUXY*beta_i
NUXY = 3*beta_i/(3+6*beta_i)
NUXY = beta_i / (1+2*beta_i)

For example, if beta_i = 0.5, then NUXY can be calculated as being 0.25.