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

محمد توفيق أحمد عراقي

2013/2014

Pb

Given 𝜶𝒇 ,𝜶𝒎 for composite materials calculate 𝜶𝟏 ,𝜶𝟐

Solution

# Main assumptions:

1. 𝝐𝟏 = 𝝐𝒇 = 𝝐𝒎

2. 𝝈𝟐 = 𝝈𝒇 = 𝝈𝒎

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

𝛼𝐹𝐿𝑓𝑇 −𝑅𝐿𝐹

𝐸𝐹𝐴𝐹−

𝑅𝐿𝑀

𝐸𝑀𝐴𝑀− 𝛼𝑀𝐿𝑀𝑇 = 0

𝑅 = 𝛼𝐹𝐿𝑓 − 𝛼𝑀𝐿𝑀 𝑇

𝐿𝑓𝐸𝐹𝐴𝐹

+𝐿𝑀

𝐸𝑀𝐴𝑀

Δ𝐿 =

𝐿𝑀

𝐸𝑀𝐴𝑀 𝛼𝐹𝐿𝑓 − 𝛼𝑀𝐿𝑀 𝑇

𝐿𝑓𝐸𝐹𝐴𝐹

+𝐿𝑀

𝐸𝑀𝐴𝑀

+ 𝛼𝑀𝐿𝑀𝑇

Applying the first assumption

𝝐𝟏 =Δ𝐿

𝐿=

𝑇(𝛼𝐹

𝐸𝑀𝐴𝑀+

𝛼𝑀

𝐸𝐹𝐴𝐹)

1𝐸𝑀𝐴𝑀

+1

𝐸𝐹𝐴𝐹

= 𝛼1 𝑇

Where

𝜐𝑀 =𝐴

𝐴𝑀 , 𝜐𝐹 =

𝐴

𝐴𝐹

Then

𝛼1 =𝜐𝐹𝐸𝐹𝛼𝐹 + 𝜐𝑀𝐸𝑀𝛼𝑀

𝜐𝐹𝐸𝐹 + 𝜐𝑀𝐸𝑀

𝛥𝐿 = 𝛥𝐿𝐹 + 𝛥𝐿𝑀 = 𝛼𝐹𝐿𝑓2 + 𝛼𝑀𝐿𝑀2 𝑇

Where subscript 2 refers to the length in the transverse direction normal to fibers and tangent to fiber

matrix plane

𝝐𝟐 =𝛥𝐿𝑀+𝛥𝐿𝐹

𝐿𝑓 + 𝐿𝑀=

𝛼𝐹𝐿𝑓2 + 𝛼𝑀𝐿𝑀2 𝑇

𝐿𝑓 + 𝐿𝑀= 𝛼2 𝑇

𝛼2 =𝛼𝐹𝑉𝐹+𝛼𝑀𝑉𝑀