To calculate the thermal expansion coefficients (α1 and α2) for a lamina with a fiber and matrix, the following steps are taken: 1. The total strain is calculated as the sum of the mechanical strain from an applied load and the thermal strain from a change in temperature. 2. Equations are developed relating stress, strain, and temperature change for loads applied in the 1 and 2 directions. 3. The equations are set equal to determine expressions for α1 and α2 in terms of the fiber and matrix thermal expansion coefficients, volumes, and moduli. 4. It is shown that α1 = (αfEfVf + αmEmVm)/(EfVf +

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