3) The strength of a titanium piece is found to be 65,000 psi when the grain size is 17 x106 m and 82,000 psi when the grain size is 0.8 x10 m. What would be the (a) ( 9- Constants in the Hall-Petch equation Strength of this titanium piece when the grain size is reduced to 0.2 x10 m.

4-52

Consider a 50 mm × 50 mm × 50 mm element of graphite-reinforced material that is heated 50°C above some reference state and is restrained in the 2 direction. (a) What are the changes in dimensions of the heated element in the 1 and 3 directions? (b) What stress 02 is required to restrain the element against deformation in the 2 direction? Assume the material properties are independent of temperature. (c) How do the changes in dimensions of the heated element for this partially restrained case compare numerically with the changes in dimensions of the heated element for the case of free thermal strain, equation (2.101)? Why are they different? (d) What are the mechanical strains for this case? Numerically compare the mechanical strains with the fully constrained case, that is, equation (2.119). Hint: To solve the problem use either equation (2.107) or (2.108), assume &₁# 0 and 83 ‡ 0, and solve for 81, 83, and 02.

How does strain hardening and grain size reduction affect elastic modulus