In our previous posts we discussed three-dimensional and two-dimensional plane strain/axisymmetric constitutive elasticity equations. In this post we will consider plane stress. This case doesn’t appear very much in geotechnical engineering; it’s more of interest to mechanical engineers and deals with problems such as this.
It’s probably a good idea to look again at the graphic for this:
The conditions of plane stress are thus:
If we compare these to plane strain, we find out that we switch the z-strain for the z-stress. This means that deriving the equations–especially the 3 x 3 case–is the mirror image of the plane strain case, and we will see this clearly below.
Let’s start by noting that, as before, the starting 4 x 4 constitutive matrix is the same as plane strain, thus
and its inverse
The stress and strain vectors, however, are
We see Equation (1a) applied to the stress vector to yield Equation (4). Multiplying through as we did before,
If we equation the two third row elements of the vectors in Equation (6), we have
Solving for the z-axis strain,
Since the z-axis strain is a function of the x-axis and y-axis strains, the obvious question is this: can we go to a 3 x 3 matrix formulation? The answer is yes if we literally invert the process. We need to do the following:
- Consider the basic equation in this form: .
- Since , we can remove the third column and row from the constitutive matrix and the third row from the stress (and strain) matrix.
- We multiply through to yield
We then invert the inverse to obtain the 3 x 3 constitutive matrix. We use the forward formulation and multiply through to obtain
After these things we use Equation (9) to obtain the z-axis strain.
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