Chapter 8, Problem 38P

The general form for a three-dimensional stress field is given by

$\left[\begin{array}{ccc}{\sigma }_{xx}& {\sigma }_{xy}& {\sigma }_{xz}\\ {\sigma }_{xy}& {\sigma }_{yy}& {\sigma }_{yz}\\ {\sigma }_{xz}& {\sigma }_{yz}& {\sigma }_{zz}\end{array}\right]$

where the diagonal terms represent tensile or compressive stresses and the off-diagonal terms represent shear stresses. A stress field $\left(\text{in MPa}\right)$ is given by

$\left[\begin{array}{ccc}10& 14& 25\\ 14& 7& 15\\ 25& 15& 16\end{array}\right]$

To solve for the principal stresses, it is necessary to construct the following matrix $\left(\text{again in MPa}\right)$:

$\left[\begin{array}{ccc}10-\sigma & 14& 25\\ 14& 7-\sigma & 15\\ 25& 15& 16-\sigma \end{array}\right]$

${\sigma }_1,-{\sigma }_2,\text{ and }{\sigma }_3$ can be solved from the equation

$\begin{array}{l}{\sigma }^2-I{\sigma }^2+II\sigma -III=0\\ \text{where }\\ \text{I=}{\sigma }_{xx}+{\sigma }_{xy}+{\sigma }_{zz}\\ II={\sigma }_{xx}{\sigma }_{yy}+{\sigma }_{xx}{\sigma }_{zz}+{\sigma }_{yy}{\sigma }_{zz}-{\sigma }_{xy}^2-{\sigma }_{xz}^2-{\sigma }_{yz}^2\\ III={\sigma }_{xx}{\sigma }_{yy}{\sigma }_{zz}-{\sigma }_{xx}{\sigma }_{yz}^2-{\sigma }_{yy}{\sigma }_{zz}-{\sigma }_{zz}{\sigma }_{xy}+2{\sigma }_{xy}{\sigma }_{xz}{\sigma }_{yz}\end{array}$

I, II, and III are known as the stress invariants. Find ${\sigma }_1,{\sigma }_2\text{ and }{\sigma }_3$ using a root-finding technique.