The Poisson's ratio of fluid lipid membranes is exactly $\nu=0.5$, because fluids can flow. As answered in this question, under the assumption of a finite bulk modulus $K$, this is the value of $\nu$ needed for a shear modulus of zero.
However, fluid lipid membranes are commonly said to be able to be stretched, with a corresponding area stretch modulus penalizing the areal strain, $A/A_0-1$. This appears fine with $\nu=0.5$ since the membrane is not necessarily incompressible even though $\nu=0.5$. However, this would also imply a nonzero Young's modulus, $E>0$, corresponding to the area stretch modulus. In contrast, the equations of linear elasticity relate $E$ to $K$ as $E=3K(1-2\nu)$, which predicts $E=0$. What is the explanation for this discrepancy?
As I see it there are two possible explanations:
Fluid lipid membranes do not really have $\nu=0.5$, but some close value. However, this would give rise to the question of what would happen for an ideal membrane for which $\nu=0.5$.
The equations of linear elasticity do not entirely apply to fluid membranes, which exhibit both elasticity and fluidity.
