SME
Aerospace Structures
Composites
Use this for “Carbon fibers (230 GPa) fill 60% of an epoxy (3.5 GPa) lamina. Find its modulus along the fibers.”
Diagram
Along: 230 × 0.6 + 3.5 × 0.4 = 139.4 GPa
This is an example. Type your own values to start a new problem.
E₁ (GPa)
=
Ef (GPa)
×
Vf
+
Eₘ (GPa)
×
Vₘ
Formulas
Vₘ = 1 − Vf
0.4 = 1 − 0.6
E₁
=
Ef
×
Vf
+
Eₘ
×
Vₘ
139.4 = 230 × 0.6 + 3.5 × 0.4
Assumptions
•
Fibers and matrix are bonded perfectly, with no voids.
•
Along the fibers both stretch the same (iso-strain), like springs side by side.
•
Each stays linear elastic, and all the fibers run one way.
Step-by-step
WE KNOW
Ef = 230 GPa, Eₘ = 3.5 GPa, Vf = 0.6
FIND
matrix volume fraction (Vₘ), modulus along the fibers (E₁)
Step 1 · Find matrix volume fraction (Vₘ)
Use Vₘ = 1 − Vf
With no voids, the matrix fills what the fibers leave.
Vₘ = 1 − 0.6
Vₘ = 0.4
Step 2 · Find modulus along the fibers (E₁)
Use E₁ = Ef × Vf + Eₘ × Vₘ
Along the fibers both stretch alike, so their stiffnesses add by share (springs side by side).
E₁
=
230
×
0.6
+
3.5
×
0.4
E₁
=
138
+
1.4
E₁
=
139.4
GPa
Check
Put the numbers back in. Both sides should match.
0.4 = 1 − 0.6 ✓
139.4 = 230 × 0.6 + 3.5 × 0.4 ✓
Topic check
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More problem types
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