A composite material is made from two or more phases. A common example is a fiber-reinforced polymer, where
strong fibers carry much of the load and a matrix binds the fibers together.
1. Volume fractions
For a two-phase composite,
\[
V_f+V_m=1,
\]
where \(V_f\) is the fiber volume fraction and \(V_m\) is the matrix volume fraction.
\[
\boxed{
V_m=1-V_f
}.
\]
2. Parallel model: longitudinal loading
If continuous fibers and matrix are loaded in the same direction, they experience approximately the same strain.
This is called the parallel model or Voigt estimate.
\[
\boxed{
E_{\parallel}=V_fE_f+V_mE_m
}.
\]
This is often a good first estimate for the longitudinal modulus of aligned continuous-fiber composites.
3. Series model: transverse loading
If the phases are loaded one after the other in the loading direction, they carry the same stress. This gives the
series model or Reuss estimate:
\[
\boxed{
\frac{1}{E_{\perp}}
=
\frac{V_f}{E_f}
+
\frac{V_m}{E_m}
}.
\]
Therefore,
\[
\boxed{
E_{\perp}
=
\frac{1}
{
\frac{V_f}{E_f}
+
\frac{V_m}{E_m}
}
}.
\]
4. Why are the two values different?
The parallel estimate is usually much larger than the series estimate when fibers are much stiffer than the
matrix. This means the material is anisotropic: its stiffness depends on direction.
\[
\text{anisotropy ratio}
=
\frac{E_{\parallel}}{E_{\perp}}.
\]
5. Shear modulus estimate
If Young’s modulus and Poisson’s ratio are known for each phase, the shear modulus of each phase can be estimated by
\[
G=\frac{E}{2(1+\nu)}.
\]
Then the same parallel and series rules can be applied:
\[
G_{\parallel}=V_fG_f+V_mG_m,
\]
\[
\frac{1}{G_{\perp}}
=
\frac{V_f}{G_f}
+
\frac{V_m}{G_m}.
\]
6. Bulk modulus estimate
For isotropic phases,
\[
K=\frac{E}{3(1-2\nu)}.
\]
A simple upper estimate is
\[
K_{\parallel}=V_fK_f+V_mK_m,
\]
and a simple lower estimate is
\[
\frac{1}{K_{\perp}}
=
\frac{V_f}{K_f}
+
\frac{V_m}{K_m}.
\]
7. Density estimate
Composite density is usually estimated by a volume-weighted average:
\[
\boxed{
\rho_c=V_f\rho_f+V_m\rho_m
}.
\]
8. Strength estimate
A very simplified parallel tensile strength estimate is
\[
\boxed{
\sigma_{\parallel}
=
V_f\sigma_f+V_m\sigma_m
}.
\]
Real strength is more complicated than stiffness. It depends strongly on fiber length, alignment, interface
bonding, voids, defects, stress concentrations, and failure mode.
9. Typical material values
| Phase |
Typical Young’s modulus |
Typical density |
Comment |
| Glass fiber |
\(\approx70\ \mathrm{GPa}\) |
\(\approx2550\ \mathrm{kg/m^3}\) |
Common and relatively economical reinforcement |
| Carbon fiber |
\(\approx230\ \mathrm{GPa}\) |
\(\approx1800\ \mathrm{kg/m^3}\) |
High stiffness-to-weight ratio |
| Aramid fiber |
\(\approx130\ \mathrm{GPa}\) |
\(\approx1440\ \mathrm{kg/m^3}\) |
High toughness and low density |
| Epoxy matrix |
\(\approx3\ \mathrm{GPa}\) |
\(\approx1200\ \mathrm{kg/m^3}\) |
Common polymer matrix |
| Polyester resin |
\(\approx3.5\ \mathrm{GPa}\) |
\(\approx1350\ \mathrm{kg/m^3}\) |
Common lower-cost matrix |
10. Worked example
Suppose a composite has
\[
V_f=0.40,
\qquad
E_f=70\ \mathrm{GPa},
\qquad
E_m=3\ \mathrm{GPa}.
\]
Then
\[
V_m=1-V_f=1-0.40=0.60.
\]
The parallel estimate is
\[
E_{\parallel}=V_fE_f+V_mE_m.
\]
\[
E_{\parallel}
=
(0.40)(70)+(0.60)(3).
\]
\[
E_{\parallel}=28+1.8=29.8\ \mathrm{GPa}.
\]
The series estimate is
\[
E_{\perp}
=
\frac{1}{\frac{0.40}{70}+\frac{0.60}{3}}.
\]
\[
E_{\perp}
\approx
4.86\ \mathrm{GPa}.
\]
Therefore,
\[
\boxed{
E_{\parallel}\approx29.8\ \mathrm{GPa}
},
\qquad
\boxed{
E_{\perp}\approx4.86\ \mathrm{GPa}
}.
\]
Key idea: aligned fibers can greatly increase stiffness along the fiber direction, but transverse stiffness remains much closer to the matrix-dominated value.
