Final answer:
The longitudinal modulus of a polymer-matrix composite with a fiber volume fraction of 47% consisting of e-glass fibers and polyester resin is approximately 36 GPa, calculated using the rule of mixtures.
Step-by-step explanation:
To calculate the longitudinal modulus of a polymer-matrix composite, which in this case consists of e-glass fibers embedded in a polyester resin matrix, we use the rule of mixtures. The rule of mixtures provides an approximation for the properties of composite materials based on the properties of its constituents and their volume fractions.
The formula for calculating the longitudinal modulus Ec is given by:
Ec = Vf ⋅ Ef + Vm ⋅ Em
where
- Vf is the volume fraction of the fibers,
- Ef is the modulus of the fibers,
- Vm is the volume fraction of the matrix (1 - Vf), and
- Em is the modulus of the matrix.
In this question, Vf (fiber volume fraction) is 47%, or 0.47, Ef (modulus of the fiber) is 73 GPa, and Em (modulus of the matrix) is 3.2 GPa. The volume fraction of the matrix, Vm, is therefore 1 - 0.47 = 0.53.
Plugging these values into our formula, we get:
Ec = 0.47 ⋅ 73 GPa + 0.53 ⋅ 3.2 GPa
Ec = 34.31 GPa + 1.696 GPa
Ec = 36.006 GPa
The longitudinal modulus of the composite is therefore approximately 36 GPa.