Chapter 2.9, Problem 73P

In many situations it is known that the normal stress in a given direction is zero. For example, ϭ_z = 0 in the case of the thin plate shown. For this case, which is known as plane stress, show that If the strains ∈_x and ∈_y have been determined experimentally, we can express ϭ_x, ϭ_y and ∈_z as follows:

$\begin{array}{l}{\sigma }_x=E\frac{{\in }_x+\nu {\in }_y}{1-{\nu }^2}\\ {\sigma }_y=E\frac{{\in }_y+\nu {\in }_x}{1-{\nu }^2}\\ {\in }_z=-\frac{\nu }{1-\nu }\left({\in }_x+{\in }_y\right)\end{array}$

Fig. P2.73