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

= ( – .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 𝛼𝑓 + 𝑉𝑚 𝐸𝑚 𝛼𝑚

𝐸

𝜎

𝜎

From equations 1,8:

+ .ΔT = (

+ .ΔT). + (

+ .ΔT).

+ .ΔT =

+ .ΔT +

+ .ΔT …. Equ(9)

From equations 7,9:

+ .ΔT =

+

) + + )

So,

𝛼 = 𝛼𝑚 𝑉𝑚 + 𝛼𝑓 𝑉𝑓