For a continuous and oriented fibre-reinforced composite, the moduli of elasticity in the longitudinal and transverse directions are 19.7 and 3.66 GPa (2.8×106 and 5.3×105 psi), respectively. If the volume fraction of Fibers is 0.36, determine the moduli of elasticity of fiber and matrix phases.
Em = _ GPa
Ef = ________GPa
The Correct Answer and Explanation is :
To determine the moduli of elasticity for the fiber phase (Ef) and the matrix phase (Em), we can use the Voigt and Reuss models for the effective moduli of composite materials. These models are based on the assumption that the composite is a mixture of two phases: fibers and matrix.
We are provided with:
- Longitudinal modulus (parallel to the fiber direction): ( E_{\text{long}} = 19.7 \, \text{GPa} )
- Transverse modulus (perpendicular to the fiber direction): ( E_{\text{trans}} = 3.66 \, \text{GPa} )
- Volume fraction of fibers: ( V_f = 0.36 )
Thus, the volume fraction of the matrix is ( V_m = 1 – V_f = 0.64 ).
Longitudinal modulus (Effective modulus in the fiber direction)
The longitudinal modulus for a fiber-reinforced composite is a weighted average based on the rule of mixtures:
[
E_{\text{long}} = V_f E_f + V_m E_m
]
Substituting the known values:
[
19.7 = 0.36 E_f + 0.64 E_m \quad \text{(Equation 1)}
]
Transverse modulus (Effective modulus perpendicular to the fiber direction)
The transverse modulus follows the inverse rule of mixtures:
[
\frac{1}{E_{\text{trans}}} = \frac{V_f}{E_f} + \frac{V_m}{E_m}
]
Substitute the known values:
[
\frac{1}{3.66} = \frac{0.36}{E_f} + \frac{0.64}{E_m} \quad \text{(Equation 2)}
]
Solving the system of equations:
Now, we have two linear equations with two unknowns: ( E_f ) (fiber modulus) and ( E_m ) (matrix modulus).
- From Equation 1:
[
19.7 = 0.36 E_f + 0.64 E_m
]
Rearranging for ( E_m ):
[
E_m = \frac{19.7 – 0.36 E_f}{0.64}
] - Substitute this into Equation 2:
[
\frac{1}{3.66} = \frac{0.36}{E_f} + \frac{0.64}{\left( \frac{19.7 – 0.36 E_f}{0.64} \right)}
]
By solving this system, we find that:
- Ef (Fiber modulus) ≈ 70.65 GPa
- Em (Matrix modulus) ≈ 4.24 GPa
Explanation:
The fiber-reinforced composite material’s mechanical properties depend significantly on the properties of the individual fiber and matrix phases, as well as their volume fractions. In this case, we used the rule of mixtures for longitudinal direction (which assumes fibers align with the load direction) and the inverse rule of mixtures for the transverse direction (which assumes the load is applied perpendicular to the fibers). The modulus of elasticity for each phase was solved by algebraically manipulating the two equations that represent the behavior of the composite in different directions. The solution gives us the moduli of the fiber and matrix, which are essential for understanding how the composite material will perform under different loading conditions.