Conversion of elastic moduli for plane stress case

Question

For isotropic material an elastic modulus, like Young's modulus and in combination with a Poisson ratio can be used to convert to any other elastic modulus, like for example the bulk modulus. Tables for these conversions exist, for example, on the wiki page of bulk modulus.

I'd like to use the conversions for FEM material models that are plane strain, plane stress or axisymmetric. What confuses me on the wiki page is that there is a mention of 3D material and 2D materials and these have different conversion formulae. It's unclear to me what 2D material means in this context. My questions:

1. Is it correct to use the 3D material conversion formulae for the plane stress case? My current thinking is that the conversions are valid for the plane strain and axisymmetric case as they model objects with 3D extend. But the plane stress case a models a thin object. Which formula should be used in that case?

2. Can anyone share a reference to where these conversions are derived, possibly including the 2D material conversions.

Answer 1

Simplified 2D analyses of 3D situations

Plain stress, plain strain and axisymmetric allow describing a domain in 2D, but it still represents a full 3D object. Therefore the material properties like Young's modulus and Poisson ratio are still the same as in 3D.

These 2D simplifications are "coded" in stress and strain tensors, which have some components equal to zero (or constant).

Relation between $E$, $\nu$ and $K$

Relation between Young's modulus $E$, Poisson ratio $\nu$ and bulk modulus $K$ in 3D comes from Hooke's law:

$$\epsilon_x = \frac{1}{E}\left(\sigma_x-\nu\sigma_y-\nu\sigma_z\right)$$ $$\epsilon_y = \frac{1}{E}\left(-\nu\sigma_x+\sigma_y-\nu\sigma_z\right)$$ $$\epsilon_z = \frac{1}{E}\left(-\nu\sigma_x-\nu\sigma_y+\sigma_z\right)$$

Now the volumetric change per volume unit $dV/V$ for small deformations is just a sum of normal strains, which may be simplified using average normal stress $\bar{\sigma}$ (~pressure):

$$\frac{dV}{V} = \epsilon_x+\epsilon_y+\epsilon_z = \left(\sigma_x+\sigma_y+\sigma_z\right)\cdot\frac{1-2\nu}{E} = 3\bar{\sigma}\frac{1-2\nu}{E}$$

Lastly, the bulk modulus as a ratio between pressure and volumetric change per unit is:

$$K = \frac{\bar{\sigma}}{dV/V} = \frac{E}{3\left(1-2\nu\right)}$$

2D materials

"2D material" relations from table at the end of wiki/Bulk_modulus may be useful for very special applications like single atom layers, where tangential stresses should not cause contraction in normal direction to the layer.