1. Home Work Solution #3
From hooks law in 1D stress state :
€m= б/E
Thermal strain:
€T=α.∆T
The total strain :
€=€m+€T
For the direction 1:
€1=€f=€m
From €= б/E+ α.∆T
€f= бf/Ef+ αf.∆T €m= бm/Em+ αm.∆T
Бf=Ef( €f- αf.T) (1)
Бm=Em( €m- αm.T) (2)
Б1==Vf*бf+Vm*бm (3)
Б1=E1( €1- α1.T) (4)
From the above equation :
Б1=Vf*Ef( €f- αf.∆T)+ Vm*Em( €m- αm.∆T)
From (4)
E1( €1- α1.T)= Vf*Ef( €f- αf.∆T)+ Vm*Em( €m- αm.∆T)
(E1* €1)-(α*E1*.∆T)=Vf*Ef*€f+ Vm*Em*€m-∆T(Vm*Em*αm+ Vm*Em*αm)
α1*E1=Vf*Ef* αf+Vm*Em*αmft #

2. б2
In the direction 2 :
Б2= бm= бf б2
€2=€m*Vf+€m*Vm
(Б2/E2)+( α2.∆T)=( бf/Ef+ αf.∆T )*Vf+( бm/Em+ αm.∆T)*Vm
(Б2/E2)+( α2.∆T)= б2*((Vf/Ef)+(Vm/Em)) +∆T(αf*Vf+αm*Vm)
α2=αf*Vf+αm*Vm #