$I_3=b_1{b}_2{b}_3=$

Example $1\mathrm{.}7.$ Three$-$Dimensional Stress in a Machine Component

The stress tensor at a point in a machine element with respect to a Cartesian coordinate system is given by the following array:

$(r)|||=\left[\begin{array}{ccc}6x& 10& 0\\ 10& 20& 40\\ 0& 40& 30\end{array}\right]$MPa

Determine the state of stress and $I_1,I_2,$ and $I_3$ for an $x^{\text{'}},y^{\text{'}},z^{\text{'}}$ coordinate system defined by rotating $x,y$ through an angle of $=45$ counterclockwise about the $z$ axis $($Fig$.1\mathrm{.}21$a$).$

Solution

First, the transformation matrix must be established:

$\{\begin{array}{ccc}\text{:}[x,& y,& z]\\ x^{\text{'}},& \frac{1}{\sqrt[\phantom{2}]{2}},\frac{1}{\sqrt[\phantom{2}]{2}},& 0\\ y,& -\frac{1}{\sqrt[\phantom{2}]{2}},\frac{1}{\sqrt[\phantom{2}]{2}},& 0\\ z^{\text{'}},& 0,0,& 1\end{array}$ ansle $(cos)$

$($a$)$

$($b$)$

Then, Eqs. $($A$)-($F$)$ of $3$D stress transformation or its matrix form must be used: