1- Derivation of thermal coefficient for composite material having a stress in the fiber direction

∈f =¿∈m=¿∈1 ¿¿

A f∗σ f+Am∗σm=A total∗σ 1F f+Fm=F total∆ x∗A f∗σ f+∆ x∗Am∗σm=A total∗σ 1 ÷∆ x∗A total

V f∗σ f+V m∗σ m=σ 1∈= σE

+α∗T↔σ=E∗(∈−α∗T )

V f∗E f (∈f−α f∗T )+V m∗Em (∈m−αm∗T )=σ 1 ∈1 (V f∗E f+V m∗Em )−T (V f∗Ef∗α f+V m∗Em∗αm )=σ1 σ 1=E1 (∈1−α1∗T )∴E1=V f∗E f+V m∗Em

∴α1=V f∗E f∗α f+V m∗Em∗αm

V f∗E f+V m∗Em

2- Derivation of thermal coefficient for composite material having a stress perpendicular to the fiber direction

σ f=σm=σ2∈f∗lf+∈m∗lm=∆l=∈2∗(l¿¿ f +lm)¿ ÷(l¿¿ f+ lm)¿ ∈2=∈f∗V f+∈m∗V m

∈2=V f ( σ fEf +α f∗T )+V m( σmEm+αm∗T )∈2=σ 2(V f∗Em+V m∗E f

E f∗Em )+T (V f∗α f+V m∗αm )

∈2=σ2E2

+α2∗T ∴E2=Ef∗Em

V f∗Em+V m∗E f∴α2=V f∗α f+V m∗αm