PSI - Issue 52

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

137

5

  , i       t n   i j ij ijk k j

( ) ( c    

c

)

 x x x

,

i

x



c

j

: i k j ijk n     .

(18)

3. Finite element method formulation The governing equations for the electric potential, temperature and elastic displacements are derived above. Now, we should derive the FEM equations. It is convenient to replace the governing equations by the weak form with lower order derivatives. The weak form of the governing equation (1) for the electric potential is given as

, H J dV H d        . i i V

(19)

Similarly, one can write the weak form for the transient heat conduction equation (8)   , , i i ik ik i i V V P m c dV d P pd E J dV                      .

(20)

The elastic problem is represented by the weak-form of equation (15)   , , ij i j ijk i jk i i i i V t R u u dV t u d R s d                . (21) Higher order partial derivatives in the weak forms (20) and (21) do not allow to apply the standard C 0 continuous interpolation of primary fields in elements. The mixed FEM is developed here, where C 0 continuous interpolation is applied independently for both displacements and strains and similarly for temperature and temperature gradients. The kinematic constraints between them are satisfied by collocation at selected internal points of elements (Tian et al. 2021). The standard C 0 continuous approximations for all primary fields are expressed in terms of nodal values and shape functions   1 2 , u u    u N q ,   1 2 ,       N q ,   1 2 ,       N q , (22) where symbols u q ,  q and  q are used for corresponding nodal values. From equations (22) one can get easily approximations for strains, normal derivative of displacements, electric intensity vector and gradients of temperature: (23) The additional independent approximations of strains and temperature gradients are needed in the mixed FEM:   1 2 ˆ , In    ε A α ,   1 2 ˆ , In    ψ A χ , (24) where  and χ represent unknown coefficients. For a simple 4-node element with a linear approximation the polynomial function matrix is given by     1 2 1 2 1 2 , 1       A . (25) The coincidence of two independent approximations (23) and (24) at selected internal points 1 2 ( , ) c c c    ξ , is leading to get unknown  coefficients  and χ 1 ( ) ( ) c c u u   α A ξ B ξ q , 1 ( ) ( ) c c     χ A ξ B ξ q . (26) Then, the independent strain and temperature gradient approximations are given by 11 1 1   22                       ε 2 1 2 ( , )   B q , 2 12 2 1 0 0 2 u u u u                         s  1 1 1 2 2 1 2 ( , ) s u   B q 2 u n n u ,1       1       1        1 2 ( , )   B q , ,2 2 2 E E          ,1          1 2 ( , )     B q . ,2  ψ