The mechanical properties of a metal may be improved by incorporating fine particles of its oxide. Given that the moduli of elasticity of the metal and oxide are, respectively, 60 GPa and 390 GPa, what is the (a) upper-bound, and (b) lower-bound modulus of elasticity values (in GPa) for a composite that has a composition of 35 vol% of oxide particles.

Uper Bound = 175.5 GPa, Lower Bound = 85.26 GPa

Explanation:

Rule of mixtures equations:

For a two-phase composite, modulus of elasticity upper-bound expression is,

E (c) (u) = E (m) x V (m) + E (p) x V (p)

Here, E (m) is the modulus of elasticity of matrix, E (p) is the modulus of elasticity of patriciate phase, E (c) is the modulus of elasticity of composite, V (m) is the volume fraction of matrix and V (p) is the volume fraction of composite.

For a two-phase composite, modulus of elasticity lower-bound expression is,

E (c) (l) = E (m) x E (p) / V (m) x E (p) + V (p) x E (m)

Consider the expression of rule of mixtures for upper-bound and calculate the modulus of elasticity upper-bound.

E (c) (u) = E (m) x V (m) + E (p) x V (p), (1)

Calculate the volume fraction of matrix.

V (m) + V (p) = 1

Substitute 0.35 for V (p).

V (m) + 0.35 = 1

V (m) = 0.65

From equation (1);

Substitute 60 GPa for E (m), 390GPa for E (p), 0.65 for V (m) and 0.35 for V (p).

E (c) (u) = E (m) x V (m) + E (p) x V (p)

E (c) (u) = (60 * 0.65) + (390 * 0.35)

E (c) (u) = 175.5 GPa

The modulus of elasticity upper-bound is 175.5GPa.

The modulus of elasticity of upper-bound can be calculated using the rule of mixtures expression. Since the sum of volume fraction of matrix and volume fraction of composite is equal to one V (m) + V (p) = 1. Substitute the value of volume fraction of matrix as 0.69 and obtain the volume fraction of matrix.

Consider the expression of rule of mixtures for lower-bound and calculate the modulus of elasticity upper-bound.

E (c) (l) = (E (m) x E (p)) / (V (m) x E (p) + V (p) x E (m))

Substitute 60 GPa for E (m), 390GPa for E (p), 0.65 for V (m) and 0.35 for V (p).

E (c) (l) = 60 * 390 / (0.65 * 390) + (0.35 * 60)

E (c) (l) = 23400/274.5

E (c) (l) = 85.26 GPa

The modulus of elasticity lower-bound is 85.26 GPa.