6.16

s = s_i + a G b (r)^1/2 (1)

Given: r = C / d (2)

where C = constant and d = grain size.

Substituting (2) into (1) gives:

s = s_i + a G b (C / d)^1/2 (3)

which is s = s_i + a G b C^1/2 (d)^-1/2

and is the same form as the Hall-Petch type equation:

s = s_i + k (d)^-1/2 where k = a G b C^1/2 Q.E.D.

[cf. pp.274-275]

6.18

Given:

s = 25 + 200 e ^0.5 for which s₀ = 25 MPa and K = 200 MPa [Holloman-Ludmick eq.]

Find: Energy of deformation/unit volume corresponding to uniform strain, e_u.

U = ò s de + constant [ the definite intergral is from e = 0 to e = e_u .

U = s₀ e₁ + (K e₁ⁿ⁺¹ ) / 2 = 25MPa (0.5) + (200MPa 0.5^1.5 ) / 2 because at e_u, e_u = n.

U = (25 + 35.4) · 10⁶ N/m² = 6 · 10⁷ J/ m²