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:
0.4
e=[0.00485 -0.00159 0.00106 0 0 0.00394]
To find the volume fraction of fibers in the composite (Vf), we can use the rule of mixtures. The rule of mixtures states that the overall properties of a composite material can be calculated based on the properties of its individual components and their volume fractions.
In this case, we have the elastic constants of the fiber composite and we can use them to calculate Vf.
The elastic constants of the fiber composite can be related to the properties of the epoxy resin and the glass fibers as follows:
E1 = Vf * Eglass + (1 - Vf) * E
ν12 = Vf * νglass + (1 - Vf) * ν
ν13 = Vf * νglass
E2 = E3 = E
Now, let's substitute the given values into these equations:
3.3GPa = Vf * 70GPa + (1 - Vf) * 2GPa
0.3 = Vf * 0 + (1 - Vf) * 0.3
1.27GPa = Vf * 0 + (1 - Vf) * 2GPa
Simplifying these equations, we get:
3.3GPa = 70Vf + 2 - 2Vf
0.3 = 0.3 - 0.3Vf
1.27GPa = 2 - 2Vf
Rearranging the equations, we have:
3.3GPa - 2 = 68Vf
0 = -0.3Vf
1.27GPa - 2 = -2Vf
Solving these equations, we find:
Vf = (3.3GPa - 2) / 68 ≈ 0.028
Vf = 0 (Discarded as it is not a physically meaningful solution)
Vf = (1.27GPa - 2) / -2 ≈ 0.365
Therefore, the volume fraction of fibers in the composite is approximately 0.028 or 2.8%.
To find the volume fraction of fibers in the composite, we need to use the rule of mixtures. The rule of mixtures states that the effective properties of a composite material can be determined based on the volume fractions and properties of its individual constituents.
In this case, we have a transversely isotropic fiber composite, with the fiber direction aligned in the 2-direction. The plane of isotropy is the 1-3 plane.
The volume fraction of fibers (Vf) can be calculated using the following formula:
Vf = (V2 - V1) / (V2 - V1 + Vf1 - Vf2)
Where:
V2 = volume fraction of epoxy resin
V1 = volume fraction of fibers in the 2-direction
Vf1 = volume fraction of fibers in the 1-direction
Vf2 = volume fraction of fibers in the 2-direction
Now, we can calculate the volume fractions:
V2 = 1 - Vf (since the epoxy resin volume fraction and fiber volume fraction should add up to 1)
V1 = Vf * Vf1
Vf2 = Vf * (1 - Vf1)
Substituting these values into the volume fraction formula, we get:
Vf = (1 - Vf - (Vf * Vf1)) / (1 - Vf - (Vf * Vf1) + Vf * (1 - Vf1) - Vf * Vf1)
Simplifying the equation further:
Vf = 1 - Vf / (1 + Vf * (Vf1 - 1))
Now, we can substitute the given values of Vf1 (volume fraction of fibers in the 1-direction) and solve for Vf:
Vf = 1 - Vf / (1 + Vf * (0.25 - 1))
To solve this equation, we can use an iterative numerical method or solve it graphically. Assuming that Vf is a positive value less than 1, we can start with an initial guess for Vf (e.g., 0.1) and then iterate until the equation is satisfied.
Using this iterative method, we can find the value of Vf.