1. Fourth Year Composite materials
Report: Solution to Homework III
Report No: 3 Date: 18/3/2013
Submitted to: Dr. Mohammad Tawfik
Name
 Mohammad Tawfik Eraky
‫عراقي‬ ‫أحمد‬ ‫توفيق‬ ‫محمد‬
2013/2014

2. Pb
Given 𝜶 𝒇 , 𝜶 𝒎 for composite materials calculate 𝜶 𝟏 , 𝜶 𝟐
Solution
# Main assumptions:
1. 𝝐 𝟏 = 𝝐 𝒇 = 𝝐 𝒎
2. 𝝈 𝟐 = 𝝈 𝒇 = 𝝈 𝒎
Assuming that interaction force to be =R
And assuming that fibers feels tension, then the total displacement of the fiber should be equal to that
of the matrix so
𝛼 𝐹 𝐿 𝑓 𝑇 −
𝑅𝐿 𝐹
𝐸 𝐹 𝐴 𝐹
−
𝑅𝐿 𝑀
𝐸 𝑀 𝐴 𝑀
− 𝛼 𝑀 𝐿 𝑀 𝑇 = 0
𝑅 =
𝛼 𝐹 𝐿 𝑓 − 𝛼 𝑀 𝐿 𝑀 𝑇
𝐿 𝑓
𝐸 𝐹 𝐴 𝐹
+
𝐿 𝑀
𝐸 𝑀 𝐴 𝑀
Δ𝐿 =
𝐿 𝑀
𝐸 𝑀 𝐴 𝑀
𝛼 𝐹 𝐿 𝑓 − 𝛼 𝑀 𝐿 𝑀 𝑇
𝐿 𝑓
𝐸 𝐹 𝐴 𝐹
+
𝐿 𝑀
𝐸 𝑀 𝐴 𝑀
+ 𝛼 𝑀 𝐿 𝑀 𝑇
Applying the first assumption
𝝐 𝟏 =
Δ𝐿
𝐿
=
𝑇(
𝛼 𝐹
𝐸 𝑀 𝐴 𝑀
+
𝛼 𝑀
𝐸 𝐹 𝐴 𝐹
)
1
𝐸 𝑀 𝐴 𝑀
+
1
𝐸 𝐹 𝐴 𝐹
= 𝛼1 𝑇
Where
𝜐 𝑀 =
𝐴
𝐴 𝑀
, 𝜐 𝐹 =
𝐴
𝐴 𝐹
Then

3. 𝛼1 =
𝜐 𝐹 𝐸 𝐹 𝛼 𝐹 + 𝜐 𝑀 𝐸 𝑀 𝛼 𝑀
𝜐 𝐹 𝐸 𝐹 + 𝜐 𝑀 𝐸 𝑀
𝛥𝐿 = 𝛥𝐿 𝐹 + 𝛥𝐿 𝑀 = 𝛼 𝐹 𝐿 𝑓2 + 𝛼 𝑀 𝐿 𝑀2 𝑇
Where subscript 2 refers to the length in the transverse direction normal to fibers and tangent to fiber
matrix plane
𝝐 𝟐 =
𝛥𝐿 𝑀+𝛥𝐿 𝐹
𝐿 𝑓 + 𝐿 𝑀
=
𝛼 𝐹 𝐿 𝑓2 + 𝛼 𝑀 𝐿 𝑀2 𝑇
𝐿 𝑓 + 𝐿 𝑀
= 𝛼2 𝑇
𝛼2 = 𝛼 𝐹 𝑉 𝐹 + 𝛼 𝑀 𝑉 𝑀