1. HW (3)
Problem: given that:
is thermal expansion coefficient for fiber
is thermal expansion coefficient for matrix
Calculate: and in for lamina
Solution:
 From hook’s law in 1D stress state, we have two types of
strains:
1. Mechanical strain according to load /Area (σ) is
=
2. Thermal strain according to temperature change ΔT
= α.ΔT
So, we combine thermal and mechanical effects
The total strain = + α.ΔT or σ = E(ϵ – α.ΔT) …….Equ(1)
Where E is the young modulus
σ is the stress
is the strain
 Apply stress σ1 in direction (1) as following:
= = ….. Equ(2)
= . + . ….. Equ(3)
So, from Equ(1) …. = ( – .ΔT) and
= ( – .ΔT)
𝜎 𝜎

2. = ( – .ΔT)…. Equ(4)
From equations 1,2,3,4:
= . ( – .ΔT) + . ( – .ΔT) or
( – .ΔT) = . ( – .ΔT) + . ( – .ΔT)
So, – .ΔT = . + .
– ΔT.( . + . ) …. Equ(5)
From equations 2,5:
– .ΔT = . + .
– ΔT.( . + . ) …. Equ(6)
From comparison between right side and left side:
= . + .
Or
 Apply stress σ2 in direction (2) as following:
= = ….. Equ(7)
= . + . ….. Equ(8)
𝛼 =
𝑉𝑓 𝐸𝑓 𝛼 𝑓 + 𝑉𝑚 𝐸 𝑚 𝛼 𝑚
𝐸
𝜎
𝜎

3. From equations 1,8:
+ .ΔT = ( + .ΔT). + ( + .ΔT).
+ .ΔT = + .ΔT + + .ΔT ….Equ(9)
From equations 7,9:
+ .ΔT = + ) + + )
So, 𝛼 = 𝛼 𝑚 𝑉𝑚 + 𝛼 𝑓 𝑉𝑓