An isotropic epoxy resin (E=2GPa, ν=0.3) is reinforced by unidirectional glass fibers (Eglass=70GPa), aligned in the 2 direction, such that the fiber composite is transversely isotropic, with the 1-3 plane being the plane of isotropy. The elastic constants of the fiber composite are:
E1=3.3GPa,
ν13=0.25
E2=29.2GPa,
ν12=0.3
E3=3.3GPa,
G12=1.27GPa
What is the volume fraction of fibers in the composite?
Vf:
To calculate the volume fraction of fibers (Vf) in the composite, we need to use the formula:
Vf = (Vf*Ef) / (Vf*Ef + Vm*Em)
where Vf is the volume fraction of fibers, Ef is the elastic modulus of the fibers, Vm is the volume fraction of the matrix (epoxy resin), and Em is the elastic modulus of the matrix.
In this case, since the composite is transversely isotropic and the fibers are aligned in the 2-direction, we can assume that Vf is the volume fraction of fibers in the 2-direction, and Vm is the volume fraction of the matrix in the 1-3 plane (plane of isotropy).
Given the elastic constants of the fiber composite, we can determine the volume fractions as follows:
E1 = (1 - Vf) * Em + Vf * Ef
3.3 GPa = (1 - Vf) * 2 GPa + Vf * 70 GPa
By rearranging the equation, we can solve for Vf:
Vf = (3.3 GPa - (1 - Vf) * 2 GPa) / (Vf * 70 GPa)
Next, we can substitute the calculated value of Vf into the elastic constants for the matrix:
E3 = (1 - Vm) * Em + Vm * Ef
3.3 GPa = (1 - Vm) * 2 GPa + Vm * 70 GPa
Again, rearranging the equation allows us to solve for Vm:
Vm = (3.3 GPa - (1 - Vm) * 2 GPa) / (Vm * 70 GPa)
Finally, since Vf + Vm = 1 (the sum of the volume fractions should equal to 1), we can determine the volume fraction of fibers:
Vf = 1 - Vm
Substitute the calculated value of Vm into this equation to find Vf.
After obtaining the values of Vf, you can calculate the volume fraction of fibers in the composite.