# Elastic Modulus Calculation for Co-WC Composite Material

## How can we calculate the upper and lower bound estimates of the elastic modulus of a Co-WC composite material?

Given the elastic modulus of cobalt (Co) as 200 GPa and the elastic modulus of tungsten carbide (WC) as 700 GPa, what are the upper and lower bound estimates?

## Answer:

To determine the upper bound Ec(u):

Ec(u) = EmVm + EpVp

Where: - Em is the elastic modulus of cobalt - Ep is the elastic modulus of the particulate - Vm is the volume fraction of cobalt - Vp is the volume fraction of particulate

Substitute the values:

Ec(u) = 200(Vm) + 700(Vp)

To determine the lower bound Ec(l):

Ec(l) = EmEp/VmEp+VpEm

Substitute the values:

Ec(l) = 200(700)/Vm(700)+Vp(200)

Ec(l) = 1400/7Vm+2Vp

## Explanation:

Elastic modulus is a measure of a material's stiffness and ability to deform under stress. In the case of a Co-WC composite material, the elastic modulus can be estimated based on the volume fractions of the two components.

The upper bound estimate considers the maximum contribution of each component to the composite's stiffness, while the lower bound estimate takes into account the minimum contribution of each component.

By substituting the given values for Co and WC elastic moduli, as well as the volume fractions, we can calculate both the upper and lower bound estimates of the composite material's elastic modulus.

In this case, the upper bound estimate is calculated by multiplying the elastic modulus of cobalt by its volume fraction, and the elastic modulus of WC by its volume fraction, and summing the results.

On the other hand, the lower bound estimate is calculated by taking the product of the elastic moduli divided by the sum of the products of volume fractions and elastic moduli for both components.

These calculations provide a range within which the elastic modulus of the Co-WC composite material is expected to fall, based on the properties of the individual components and their volume fractions in the composite.