Plane stress, plane strain and axisymmetric

The basic relationship between stress and strain is \sigma = C \epsilon$ and the stiffness matrix $C$ is of the form

$$C = \begin{bmatrix} c_{11} & c_{12} & c_{13} & 0 & 0 & 0 \\ c_{21} & c_{22} & c_{23} & 0 & 0 & 0 \\ c_{31} & c_{32} & c_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & c_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & c_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & c_{66} \\ \end{bmatrix}$$

For plane strain the strains $\epsilon_{zz}$, $\epsilon_{yz}$ and $\epsilon_{zz}$ are zero, and we are not directly interested in the corresponding stresses so we can reduce the stiffness matrix to a 3 $\times$ 3 matrix.

$$C = \begin{bmatrix} c_{11} & c_{12} & 0 \\ c_{21} & c_{22} & 0 \\ 0 & 0 & c_{44} \\ \end{bmatrix}$$

Axisymmetric is similar to plane strain, reducing the problem to two dimensions, radial ($x$) and axial ($y$), but in this instance the strain in the third (hoop $z$) direction is related to the radial strain. In this case the strains $\epsilon_{yz}$ and $\epsilon_{zx}$ are zero, and we are not directly interested in the corresponding stresses so we can reduce the stiffness matrix to a 4 $\times$ 4 matrix.

$$C = \begin{bmatrix} c_{11} & c_{12} & c_{13} & 0 \\ c_{21} & c_{22} & c_{23} & 0 \\ c_{31} & c_{32} & c_{33} & 0 \\ 0 & 0 & 0 & c_{44} \\ \end{bmatrix}$$

For plane stress the stresses $\sigma_{zz}$, $\sigma_{yz}$ and $\sigma_{zx}$ are zero, so we can partition the stiffness matrix

$$C = \begin{bmatrix} c_{11} & c_{12} & 0 & c_{13} & 0 & 0 \\ c_{21} & c_{22} & 0 & c_{23} & 0 & 0 \\ 0 & 0 & c_{44} & 0 & 0 & 0 \\ c_{31} & c_{32} & 0 & c_{33} & 0 & 0 \\ 0 & 0 & 0 & 0 & c_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & c_{66}\\ \end{bmatrix}$$

so that the stress strain relationship can focus on the terms of interest $i$ and the terms that can be removed $r$.

$$\begin{bmatrix} \sigma_i \\ 0 \\ \end{bmatrix} = \begin{bmatrix} C_{ii} & C_{ir} \\ C_{ri} & C_{rr} \\ \end{bmatrix} \begin{bmatrix} \epsilon_i \\ \epsilon_r \\ \end{bmatrix}$$

This allows the strain corresponding to unstressed term $j$ to be removed giving the updated equation

$$\sigma_i = (C_{ii} - C_{ir} C_{rr}^{-1} C_{ri}) \epsilon_i$$

where $C_{ii} - C_{ir} C_{rr}^{-1} C_{ri}$ is the compliance matrix for plane stress.