Generalized 3D Stress

In 3D stress instead of Plane Stress, we have to deal with 3 principal directions and 3 principal stresses. This involves finding the roots of the cubic equation:

$$\begin{array}{rl}& {\sigma }^3-({\sigma }_x+{\sigma }_y+{\sigma }_z){\sigma }^2\\ & +({\sigma }_x{\sigma }_y+{\sigma }_x{\sigma }_z+{\sigma }_y{\sigma }_z-{\tau }_{xy}^2-{\tau }_{yz}^2-{\tau }_{zx}^2)\sigma \\ & -({\sigma }_x{\sigma }_y{\sigma }_z+2{\tau }_{xy}{\tau }_{yz}{\tau }_{zx}-{\sigma }_x{\tau }_{yz}^2-{\sigma }_y{\tau }_{zx}^2-{\sigma }_z{\tau }_{xy}^2)=0\end{array}$$

There are also three principal shears:

$${\tau }_{1/2}=\frac{{\sigma }_1-{\sigma }_2}{2},{\tau }_{2/3}=\frac{{\sigma }_2-{\sigma }_3}{2},{\tau }_{1/3}=\frac{{\sigma }_1-{\sigma }_3}{2}$$

• For plane stress, the stress-free surface will have one of the principal stresses equal zero.