SME
Aerospace Structures · Problem type
Modulus across the fibers
Use this for “Carbon fibers (230 GPa) fill 60% of an epoxy (3.5 GPa) lamina. Find its modulus across the fibers.”
Diagram
Across:
1
0.6 ÷ 230 + 0.4 ÷ 3.5
=
8.555
GPa
This is an example. Type your own values to start a new problem.
1
E₂ (GPa)
=
Vf
Ef (GPa)
+
Vₘ
Eₘ (GPa)
Formulas
Vₘ = 1 − Vf
0.4 = 1 − 0.6
1
E₂
=
Vf
Ef
+
Vₘ
Eₘ
1
8.555
=
0.6
230
+
0.4
3.5
Assumptions
•
Fibers and matrix are bonded perfectly, with no voids.
•
Across the fibers both carry the same stress (iso-stress), like springs in series.
•
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 across the fibers (E₂)
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 across the fibers (E₂)
Use 1 ÷ E₂ = Vf ÷ Ef + Vₘ ÷ Eₘ
Across the fibers both carry the same stress, so their flexibilities add (springs in series).
E₂
=
1
Vf ÷ Ef + Vₘ ÷ Eₘ
E₂
=
1
0.6 ÷ 230 + 0.4 ÷ 3.5
E₂
=
1
0.002609 + 0.1143
E₂
=
1
0.1169
E₂
=
8.555
GPa
Check
Put the numbers back in. Both sides should match.
0.4 = 1 − 0.6 ✓
1
8.555
=
0.6
230
+
0.4
3.5
✓
Topic check
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