A 1-cm-diameter steel ball is cooled after austenitization. At one point during the cooling, a 0.5-mm-thick layer at the surface transforms to martensite while the center is still austenite. For simplicity, assume that the interior is at an average temperature of 200?C and the surface is at 20?C and that there are no stresses in the material at this point. When the center cools to 20?C, it transforms to martensite. For this steel, the austenite-to-martensite transformation is accompanied by a 1.2% volume expansion. Assume that this expansion occurs equally in all directions. For steel, the coef?cient of linear thermal expansion is α = 6 × 10-6/?C, Young's modulus is 30 × 106 psi, and υ = 0.29. Assume these apply to both martensite and austenite. Find the stress state at the surface.