Rubber particles, roughly 1mm in diameter, have been added to an epoxy resin to increase its resistance to crack growth. Estimate the Young's modulus of the composite, given that the Young's moduli of the epoxy and the rubber are, respectively, 2GPa and 10MPa, and that the volume fraction of the rubber particles is 5%.
To estimate the Young's modulus of the composite material, we can use the rule of mixtures. The rule of mixtures is a simple mathematical model that approximates the behavior of a composite material based on the properties of its constituent materials and their volume fractions.
In this case, we have an epoxy resin and rubber particles mixed together. The volume fraction of the rubber particles is given as 5%, meaning that 5% of the composite volume is occupied by rubber particles and the rest is epoxy.
The rule of mixtures states that the effective Young's modulus (E_comp) of the composite is given by:
E_comp = V_epoxy * E_epoxy + V_rubber * E_rubber
where V_epoxy and V_rubber are the volume fractions of epoxy and rubber, and E_epoxy and E_rubber are the Young's moduli of epoxy and rubber, respectively.
Given:
- Volume fraction of rubber particles (V_rubber) = 5% = 0.05
- Volume fraction of epoxy (V_epoxy) = 1 - V_rubber = 1 - 0.05 = 0.95
- Young's modulus of epoxy (E_epoxy) = 2 GPa = 2 * 10^9 Pa
- Young's modulus of rubber (E_rubber) = 10 MPa = 10 * 10^6 Pa
Substituting these values into the equation, we get:
E_comp = 0.95 * (2 * 10^9 Pa) + 0.05 * (10 * 10^6 Pa)
Simplifying the expression:
E_comp = 1.9 * 10^9 Pa + 0.5 * 10^6 Pa
E_comp = 1.9 * 10^9 Pa + 0.5 * 10^6 Pa
E_comp = 1.95 * 10^9 Pa
Therefore, the estimated Young's modulus of the composite is approximately 1.95 GPa.