Biswajit Banerjee

### Plane stress return: Spectral decomposition

##### Introduction

The previous parts of this series dealt with:

Let us now take a slight detour and examine the spectral decomposition of the stiffness matrix ($\mathbf{C}$) and the transformation matrix ($\mathbf{P}$).

##### Spectral decomposition of $\mathbf{C}$

Recall that under plane stress conditions the stiffness matrix reduces to

$$\mathbf{C} = \begin{bmatrix} \frac{4\mu(\lambda+\mu)}{\lambda+2\mu} & \frac{2\mu\lambda}{\lambda+2\mu} & 0 \\ \frac{2\mu\lambda}{\lambda+2\mu} & \frac{4\mu(\lambda+\mu)}{\lambda+2\mu} & 0 \\ 0 & 0 & \mu \end{bmatrix}$$

Since $\mathbf{C}$ is a real, symmetric, matrix we can find a spectral decomposition of the matrix via a singular value decomposition of the form

$$\mathbf{C} = \mathbf{Q}_C \mathbf{L}_C \mathbf{Q}_C^T$$

where $\mathbf{Q}_C$ is an orthogonal matrix whose columns form the basis of the spectral decomposition and $\mathbf{L}_C$ is a diagonal matrix containing the eigenvalues of the spectral decomposition.

Because we are lazy and prone to algebra errors, let us turn to Mathematica to solve the singular value decomposition problem. The script we use is:

#### Remarks

We will use these decompositions to diagonalize the complicated matrix expressions in the next part of this series.