SME
Materials Science & Material Properties
Composite materials (rule of mixtures)
Use this for “Carbon fibers (E = 230 GPa, ρ = 1.80 g/cm³) make up 60% of an epoxy composite (E = 3.5 GPa, ρ = 1.25 g/cm³). Find E₁ and the density.”
Diagram
Along: 230 × 0.6 + 3.5 × 0.4 = 139.4 GPa
Density: 1.8 × 0.6 + 1.25 × 0.4 = 1.58 g/cm³
Specific modulus: 88.23 MN·m/kg
This is an example. Type your own values to start a new problem.
E₁ (GPa)
=
Ef (GPa)
×
Vf
+
Eₘ (GPa)
×
(1
−
Vf
)
ρc (g/cm³)
=
ρf (g/cm³)
×
Vf
+
ρₘ (g/cm³)
×
(1
−
Vf
)
E₁ ÷ ρc
Specific modulus
worked out (not typed) · MN·m/kg
Formulas
E₁
=
Ef
×
Vf
+
Eₘ
×
(1
−
Vf)
139.4 = 230 × 0.6 + 3.5 × (1 − 0.6)
ρc = ρf × Vf + ρₘ × (1 − Vf)
1.58 = 1.8 × 0.6 + 1.25 × (1 − 0.6)
E₁
ρc
=
E₁
ρc
88.23
=
139.4
1.58
Assumptions
•
Continuous aligned fibers, perfectly bonded, so fiber and matrix stretch together (isostrain).
•
No voids: the matrix fills the rest, so Vₘ = 1 − Vf.
•
Vf is a fraction from 0 to 1: 60% fiber is 0.6.
Step-by-step
WE KNOW
Vf = 0.6, Ef = 230 GPa, Eₘ = 3.5 GPa, ρf = 1.8 g/cm³, ρₘ = 1.25 g/cm³
FIND
longitudinal modulus (E₁), composite density (ρc), specific modulus (E₁ ÷ ρc)
Step 1 · Find longitudinal modulus (E₁)
Use E₁ = Ef × Vf + Eₘ × (1 − Vf)
Along the fibers both stretch alike, so their stiffnesses add by volume share (springs side by side).
E₁
=
230
×
0.6
+
3.5
×
(1
−
0.6)
E₁
=
230
×
0.6
+
3.5
×
0.4
E₁
=
138
+
1.4
E₁
=
139.4
GPa
Step 2 · Find composite density (ρc)
Use ρc = ρf × Vf + ρₘ × (1 − Vf)
Mass adds by volume share: each part’s density times its fraction.
ρc = 1.8 × 0.6 + 1.25 × (1 − 0.6)
ρc = 1.8 × 0.6 + 1.25 × 0.4
ρc = 1.08 + 0.5
ρc = 1.58 g/cm³
Step 3 · Find specific modulus (E₁ ÷ ρc)
Use E₁ ÷ ρc = E₁ ÷ ρc
Stiffness
per
unit
mass:
divide
E₁
by
the
composite’s
density.
E₁
ρc
=
139.4
1.58
E₁
ρc
=
88.23
MN·m/kg
Check
Put the numbers back in. Both sides should match.
139.4 = 230 × 0.6 + 3.5 × (1 − 0.6) ✓
1.58 = 1.8 × 0.6 + 1.25 × (1 − 0.6) ✓
88.23
=
139.4
1.58
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
More problem types
Modulus across the fibers
Use this for “Carbon fibers (230 GPa) make up 60% of an epoxy (3.5 GPa) composite loaded across the fibers. Find E₂.”
Load share and longitudinal strength
Use this for “Carbon fibers (230 GPa) make up 60% of an epoxy (3.5 GPa). What share of an axial load do the fibers carry, and what is the strength if the fibers break at 3500 MPa while the epoxy carries 40 MPa?”