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?

0.4

To determine the volume fraction of fibers in the composite, we need to use the rule of mixtures. The rule of mixtures states that the properties of a composite material can be determined by considering the weighted average of the properties of its individual constituents.

In this case, we have an epoxy matrix and glass fibers. Let's denote the volume fraction of the fiber material as Vf, and the volume fraction of the epoxy matrix as Vm. Since the composite is transversely isotropic, we can assume that the volume fractions of the fiber material are the same in the 1-2 plane (along the fiber direction) and in the 3 direction (transverse to the fiber direction).

According to the rule of mixtures, the effective Young's modulus of the composite in the 1 direction (E1_eff) can be calculated as:

E1_eff = Vf * Ef1 + Vm * Em1,

where Ef1 is the Young's modulus of the fiber material in the 1 direction, and Em1 is the Young's modulus of the matrix material in the 1 direction.

Similarly, the effective Young's modulus of the composite in the 2 direction (E2_eff) and 3 direction (E3_eff) can be calculated as:

E2_eff = Vf * Ef2 + Vm * Em2,
E3_eff = Vf * Ef3 + Vm * Em3,

where Ef2 and Ef3 are the Young's moduli of the fiber material in the 2 and 3 directions, respectively, and Em2 and Em3 are the Young's moduli of the matrix material in the 2 and 3 directions, respectively.

Since we are given the values of the elastic constants of the composite (E1, E2, and E3) and the elastic constants of the individual constituents (Ef1, Ef2, Ef3, Em1, Em2, and Em3), we can set up a system of equations using these expressions and solve for the volume fraction of the fibers (Vf).

Let's substitute the given values into the equations:

E1_eff = Vf * Ef1 + Vm * Em1,
E2_eff = Vf * Ef2 + Vm * Em2,
E3_eff = Vf * Ef3 + Vm * Em3.

E1_eff = 3.3 GPa,
E2_eff = 29.2 GPa,
E3_eff = 3.3 GPa,
Ef1 = 3.3 GPa,
Ef2 = 70 GPa,
Ef3 = 3.3 GPa,
Em1 = 2 GPa,
Em2 = ??? (not given),
Em3 = ??? (not given).

Unfortunately, we don't have the values for Em2 and Em3, which are the Young's moduli of the matrix material in the 2 and 3 directions, respectively. Without these values, it is not possible to determine the volume fraction of fibers in the composite.