PSI - Issue 5

B. Boukert et al. / Procedia Structural Integrity 5 (2017) 115–122 B.Boukert / Structural Integrity Procedia 00 (2017) 000 – 000

5

119

The mechanical transverse load q(x,y) is also expanded in double Fourrier series

1 1      n m  

q x y

sin( ) sin( ) mn Q x y  

( , )

(8)

0 0 4 ( , ) sin( ) sin( ) a b q x y x ab    

mn Q

y dxdy

(9)



The coefficients ( U mn ,V mn , W mn , X mn , Y mn ) of the Navier solution are obtained through the resolution of the system given by (10) .

 

 

T

C

1

1

N N

N

 







mn     U V

11 12 13 14 15 S S S S S

        

 

mn

mn

 

 

T

C

2

2

N





12 22 23 24 25 13 23 33 34 35 14 24 34 44 45 15 25 35 45 55 S S S S S S S S S S W Q S S S S S X S S S S S Y                         mn mn mn mn

mn

mn

(10)

    

mn

T

C

1

1

2 mn M M M M     T mn



mn

C

2







mn

Where N mn , M mn are given by equations 11-13:

1

1 k z k N z k k z k N z k          1 N z k 

    k Q 

k

Ti

N

( ) T z dz





mn

i

1



    k Q 

k

Ti

M

( ) T z zdz





(11)

mn

i

1

    k Q 

k

Ti

3

P

( ) T z z dz





mn

i

1 k z k 

4 h

Ti

Ti

Ti

mn M M 

P



(12)

mn

mn

2

3

0 0   a b

4

i

i

mn T

T

sin( ) sin( ) x  

y dxdy

(13)



ab

    

( , , ), T x y z C x y z   ( , , )

( ( , ), ( , )) sin( ) sin( ) f z T f z C x y  

(14)

n m

1 1

 

The simulated temperature ∆ T & concentration ∆ C profile are indicated in table 1.

Table 1.Temperature ∆ T & concentration ∆ C profile shape Graphe reference Mech-Transverse load f(z,T)

f(z,C)

M1[1]

Yes Yes Yes Yes Yes Yes

0

0

MTC2[11] MTC3[8] MTC4[9] MTC5[10] MTC6[17]

T 0 +ZT 1

C 0 +ZC 1

2T 0 (z/h) T 0 (1/2+z/h) T 0 (1-4z 2 /h 2 ) T 0 +T 1 (2z/h)+T 2 (4z

2C 0 (z/h) C 0 (1/2+z/h) C 0 (1-4z 2 /h 2 ) C 0 +C 1 (2z/h)+ C 2 (4z

2 /h 2 )

2 /h 2 )