SME
Materials Science & Material Properties · Problem type
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.”
Diagram
Across:
1
0.6 ÷ 230 + 0.4 ÷ 3.5
=
8.555
GPa
Along, for comparison: 139.4 GPa
This is an example. Type your own values to start a new problem.
1
E₂ (GPa)
=
Vf
Ef (GPa)
+
1
−
Eₘ (GPa)
E₁
Longitudinal modulus
worked out (not typed)
E₁ ÷ E₂
Anisotropy ratio
worked out (not typed)
Formulas
1
E₂
=
Vf
Ef
+
1 − Vf
Eₘ
1
8.555
=
0.6
230
+
1 − 0.6
3.5
E₁
=
Ef
×
Vf
+
Eₘ
×
(1
−
Vf)
139.4 = 230 × 0.6 + 3.5 × (1 − 0.6)
E₁
E₂
=
E₁
E₂
16.3
=
139.4
8.555
Assumptions
•
Across the fibers, fiber and matrix carry the same stress (isostress), like springs in series.
•
This is a lower bound: real laminae run a little stiffer across the fibers.
•
Perfect bonding and no voids; Vf is a fraction from 0 to 1.
Step-by-step
WE KNOW
Vf = 0.6, Ef = 230 GPa, Eₘ = 3.5 GPa
FIND
transverse modulus (E₂), longitudinal modulus (E₁), anisotropy ratio (E₁ ÷ E₂)
Step 1 · Find transverse modulus (E₂)
Use 1 ÷ E₂ = Vf ÷ Ef + (1 − Vf) ÷ Eₘ
Across the fibers both carry the same stress, so their flexibilities add (springs in series).
E₂
=
1
Vf ÷ Ef + (1 − Vf) ÷ Eₘ
E₂
=
1
0.6 ÷ 230 + (1 − 0.6) ÷ 3.5
E₂
=
1
0.6 ÷ 230 + 0.4 ÷ 3.5
E₂
=
1
0.002609 + 0.1143
E₂
=
1
0.1169
E₂
=
8.555
GPa
Step 2 · Find longitudinal modulus (E₁)
Use E₁ = Ef × Vf + Eₘ × (1 − Vf)
Along the fibers both stretch alike, so their stiffnesses add by volume share.
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 3 · Find anisotropy ratio (E₁ ÷ E₂)
Use E₁ ÷ E₂ = E₁ ÷ E₂
How many times stiffer the lamina is along the fibers than across them.
E₁
E₂
=
139.4
8.555
E₁
E₂
=
16.3
Check
Put the numbers back in. Both sides should match.
1
8.555
=
0.6
230
+
1 − 0.6
3.5
✓
139.4 = 230 × 0.6 + 3.5 × (1 − 0.6) ✓
16.3
=
139.4
8.555
✓
← Previous lesson
Composite materials (rule of mixtures)
Next lesson →
Load share and longitudinal strength
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.