고체역학 (Beer) — Stress and Strain—Axial Loading: Problem 2.73

2.73 In many situations it is known that the normal stress in a given direction is zero. For example, \(\sigma_z = 0\) in the case of the thin plate shown. For this case, which is known as plane stress, show that if the strains \(\epsilon_x\) and \(\epsilon_y\) have been determined experimentally, we can express \(\sigma_x\), \(\sigma_y\), and \(\epsilon_z\) as follows: \[\sigma_x = E \frac{\epsilon_x + \nu\epsilon_y}{1 - \nu^2}\] \[\sigma_y = E \frac{\epsilon_y + \nu\epsilon_x}{1 - \nu^2}\] \[\epsilon_z = -\frac{\nu}{1 - \nu} (\epsilon_x + \epsilon_y)\]