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, 68 GPa and 380 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 63 vol% of oxide particles.

a) 254.6 GPa

b) 140.86 GPa

Explanation:

a) Considering the expression of rule of mixtures for upper-bound and calculating the modulus of elasticity for upper bound;

Ec (u) = EmVm + EpVp

To calculate the volume fraction of matrix, 0.63 is substituted for Vp in the equation below,

Vm + Vp = 1

Vm = 1 - 0.63

Vm = 0.37

In the first equation,

Where

Em = 68 GPa, Ep = 380 GPa, Vm = 0.37 and Vp = 0.63,

The modulus of elasticity upper-bound is,

Ec (u) = EmVm + EpVp

Ec (u) = (68 x 0.37) + (380 x 0.63)

Ec (u) = 254.6 GPa.

b) Considering the express of rule of mixtures for lower bound;

Ec (l) = (EmEp) / (VmEp + VpEm)

Substituting values into the equation,

Ec (l) = (68 x 380) / (0.37 x 380) + (0.63 x 68)

Ec (l) = 25840/183.44

Ec (l) = 140.86 GPa