Chapter 14.11, Problem 110P

Use Hooke’s law, Eq. 14–32, to develop the strain tranformation equations, Eqs. 14–19 and 14–20, from the stress tranformation equations, Eqs. 14–1 and 14–2.

${\sigma }_{x^{\prime }}=\frac{{\sigma }_x+{\sigma }_y}{2}+\frac{{\sigma }_x-{\sigma }_y}{2}\cos 2\theta +{\tau }_{xy}\sin 2\theta$    (14–1)

${\tau }_{x^{\prime }{y}^{\prime }}=-\frac{{\sigma }_x-{\sigma }_y}{2}\sin 2\theta +{\tau }_{xy}\cos 2\theta$    (14–2)

Stress Components Acting along x′, y′ Axes

${ϵ}_{x^{\prime }}=\frac{{ϵ}_x+{ϵ}_y}{2}+\frac{{ϵ}_x-{ϵ}_y}{2}\cos 2\theta +\frac{{\gamma }_{xy}}{2}\sin 2\theta$    (14–19)

$\frac{{\gamma }_{x^{\prime }{y}^{\prime }}}{2}=-\left(\frac{{ϵ}_x-{ϵ}_y}{2}\right)\sin 2\theta +\frac{{\gamma }_{xy}}{2}\cos 2\theta$    (14–20)

Normal and Shear Strain Components

$\begin{array}{l}{ϵ}_x=\frac{1}{E}\left[{\sigma }_x-v\left({\sigma }_y+{\sigma }_z\right)\right]\\ {ϵ}_y=\frac{1}{E}\left[{\sigma }_y-v\left({\sigma }_x+{\sigma }_z\right)\right]\\ {ϵ}_z=\frac{1}{E}\left[{\sigma }_z-v\left({\sigma }_x+{\sigma }_y\right)\right]\end{array}$    (14–32)