discover modulus of elasticity of the compositea

Question

Discover modulus of elasticity of the composite:

A composite of glass fibres and epoxy contain all fibres laid along the length and is needed to carry a stress of 1200 MPa. The limiting stress carried by fibre & epoxy are respectively 2400 MPa and 80 MPa. The volume ratios and modulii of elasticity are defined below :

V _f  = 70%,    V_m = 30%,       E _f  = 72 GPa,            E_m = 3 GPa

The composite is stretched in the direction of fibres.

 (a)       Discover modulus of elasticity of the composite.

 (b)       What percentage of broken fibres might be tolerated if matrix is not to be stressed more than 80% of its capacity.

Solution

E = V _f E _f  + V_m  E_m = 0.7 × 72 + 0.3 × 3 = 51.3 MPa

In Eq.,

 σ′ = 1200 MPa, σ′m = 0.8 × 80 = 64 MPa, E f/ Em    = 72 /3 = 24

∴          x = (1200 - 64 + 0.7 (1 - 24) 64) / (1200 + 64 (1 - 24))

               = 0.388

or, 38.8% broken fibres might be tolerated.