The basic relationship between stress and strain is \sigma = C \epsilon$ and the stiffness matrix is of the form
For plane strain the strains , and are zero, and we are not directly interested in the corresponding stresses so we can reduce the stiffness matrix to a 3 3 matrix.
Axisymmetric is similar to plane strain, reducing the problem to two dimensions, radial () and axial (), but in this instance the strain in the third (hoop ) direction is related to the radial strain. In this case the strains and are zero, and we are not directly interested in the corresponding stresses so we can reduce the stiffness matrix to a 4 4 matrix.
For plane stress the stresses , and are zero, so we can partition the stiffness matrix
so that the stress strain relationship can focus on the terms of interest and the terms that can be removed .
This allows the strain corresponding to unstressed term to be removed giving the updated equation