Composite Rule of Mixtures Calculator

Rule of Mixtures (ROM) The rule of mixtures predicts properties of fiber-reinforced composites from fiber and matrix properties. Valid when fibers are continuous, aligned, and well-bonded to the matrix.

Volume Fractions Vf + Vm = 1 (fiber + matrix = 100%) Typical Vf: unidirectional prepreg 0.55 to 0.65 | hand layup GFRP 0.30 to 0.45 | woven fabric 0.45 to 0.55

Longitudinal Modulus (along fibers: iso-strain assumption) E₁ = Ef × Vf + Em × Vm ← Voigt upper bound Both fiber and matrix experience the same strain → stiffnesses add in parallel.

Transverse Modulus (perpendicular to fibers: iso-stress assumption) 1/E₂ = Vf/Ef + Vm/Em ← Reuss lower bound Both fiber and matrix carry the same stress → compliances add in series.

Density ρc = ρf × Vf + ρm × Vm (linear rule of mixtures, exact for a well-consolidated composite)

Longitudinal Tensile Strength σ₁* = σf* × Vf + σm’ × Vm Where σm’ = matrix stress at fiber failure strain. Requires Vf > Vcrit to achieve composite strengthening. Vcrit = (σm* − σm’)/(σf* − σm')

Transverse Strength and Other Properties Transverse strength is matrix-dominated, so ROM is less accurate here. Use Tsai-Hill or Halpin-Tsai. Shear modulus (Halpin-Tsai): G₁₂ = Gm(1+ξηVf)/(1−ηVf), where η=(Gf−Gm)/(Gf+ξGm), ξ≈1

Typical Fiber Properties Carbon (AS4): Ef = 235 GPa, σf = 3930 MPa, ρ = 1.79 g/cm³ Glass (E-glass): Ef = 73 GPa, σf = 3450 MPa, ρ = 2.60 g/cm³ Aramid (Kevlar 49): Ef = 124 GPa, σf = 3600 MPa, ρ = 1.44 g/cm³

Typical Matrix Properties Epoxy: Em = 3.5 GPa, σm = 70 MPa, ρ = 1.25 g/cm³ Polyester: Em = 3.0 GPa, σm = 55 MPa, ρ = 1.20 g/cm³

The quasi-isotropic laminate, and why E₁ is a trap

Almost nobody builds a part from unidirectional plies all running one way, because such a part is superb along the fibers and close to useless across them. Real structures use a balanced stack, typically [0/+45/-45/90], which trades peak stiffness for the same stiffness in every in-plane direction.

The rule of thumb for a fiber-dominated system: a quasi-isotropic laminate reaches roughly 35 to 40% of E₁. Carbon/epoxy at E₁ = 140 GPa lands near 50 GPa as a laminate, which is still a quarter of steel’s stiffness at a fifth of its weight, but it is not 140.

This matters because the number people quote from a datasheet is usually E₁. If you size a part on E₁ and then build it quasi-isotropic, it will be about a third as stiff as you expected. The calculator prints both.

Where the rule of mixtures stops working

ROM assumes continuous, aligned, perfectly bonded fibers. Reality intrudes in four places:

• Short or chopped fibers need the Halpin-Tsai equations. Below a critical fiber length the fiber cannot pick up full load through shear from the matrix, and E₁ falls well under the ROM value.
• Poor bonding at the fiber-matrix interface means the two do not share strain, and both the modulus and the strength drop.
• Voids from bad consolidation cut strength faster than they cut density. Aerospace prepreg targets under 1% void content for exactly this reason.
• Transverse and shear properties are matrix-dominated and the Reuss bound is only a floor. Real E₂ sits between Reuss and Voigt, usually nearer Reuss.