PSI - Issue 52

Jan Sladek et al. / Procedia Structural Integrity 52 (2024) 133–142 Author name / Structural Integrity Procedia 00 (2019) 000–000

138 6

1 2 ( , )    ψ A L q , ˆ In  

1 2 ( , )    ε A L q , ˆ In u 

(27)

1 ( ) ( ) c c u   L A ξ B ξ and 

1 ( ) ( ) c c   L A ξ B ξ .

where





Finally, one can write

1       2  

1       2  













ˆ η

ˆ ε

*

*

In

Ïn

1 2   A α A α A L q . 1 2   1 2   , , , u   













ˆ η A eIn 

 q ,

*

u  L q βB 

1 2   ,

1 2   ,







1 ˆ         ψ ψ Ïn

A A

1 2     ,

  





1

*

 (28) Substituting above spatial approximations into the weak forms (19)-(21) and taking into account that variations   u  q ,     q and     q can be arbitrary, we obtain system of ordinary differential equations (ODE) ( ) ( ) T T V H dV d                 B ξ SB ξ q N H , (29)   * 2 * 1 2 ( ) ( , ) ( ) ( ) T T T T V l dV                  B ξ κB L A ξ MA ξ L q  1 2   2 1 2 2 , ˆ Ïn A           L q . , In Θ χ

  

  

( ) N ξ N ξ ( ) T c 

T q N Λ B P N T d d           s

T

,

(30)

dV

QdV











V

V





P



  

  

  

 

  

  

2

* L A ξ GA ξ * ( ) ( ) T T 

T u

( ) B ξ CB ξ ( )

 L q

dV l  

dV





u

u



V

V

  

  

  

 

  

  

2

* L A ξ GβB ξ ( ) ( ) T T

T u

T q N T B R . d d   

T u

( ) B ξ γN ξ ( )

dV l  

dV



(31)









s









V

V





t

R

A constant approximation for time derivatives within the time interval 

 1 , k k    , is considered in the FEM equation

(30)

1 ( ) x

k

k

( ) x









( , ) ( , )      x x  (32) where the time interval [0,T] is discretized uniformly into W subintervals. Then, we can define time k k     ( 0,1,..., ) k W  with the time step / T W    . In the Joule heating problems the electric field is uncoupled with thermal and mechanical fields. Therefore, in the first step it is solved equation (29) to get a distribution of the electric potential and the Joule heat source. Then, the non stationary heat conduction problem is solved for the known Joule heat source. Finally, mechanical fields are solved. 3. Numerical examples In the numerical example material coefficients correspond to titanium alloy Ti-6Al-4V. A square plate is analyzed, with the geometric parameters 7 5 , 1.0 10 m w a a     . On the bottom and top surfaces there is prescribed uniform electric current density 4 2 0 10 / H A m  (see Fig. 2). A symmetry is utilized in numerical analyses. The heat flux is considered to be vanishing on the whole boundary. Furthermore, on the crack surfaces and on the right lateral side 0 H  , 1 2 0 t t   ; on the left side 0 H  , 1 2 0 u t   ; on the top side 0 H H  , 1 0 t  . The combined electromechanical load is considered to have a finite crack opening (no contact) as an initial value. After switching-      , ( ) ( , ) k k     x x ,