This report summarizes the solution to a homework problem on calculating the coefficients of thermal expansion for a composite material. 1) Main assumptions are that the strain in the fibers equals the strain in the matrix, and the stress in the fibers equals the stress in the matrix. 2) An interaction force R is assumed, and setting the total fiber displacement equal to the total matrix displacement gives an expression for R. 3) The coefficient of thermal expansion in the direction of the fibers, α1, is calculated in terms of the coefficients and properties of the fibers and matrix. 4) The coefficient of thermal expansion transverse to the fibers, α2, is similarly calculated.

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