PSI - Issue 52

Haolin Li et al. / Procedia Structural Integrity 52 (2024) 752–761 HaolinLi / Structural Integrity Procedia 00 (2023) 000–000

756

5

represent the iteration numbers:

  

ε i + 1 ( x ) = ε i ( x ) − G 0( N ) ( x ) ∗ N i ( x ) + ε ϕ i + 1 ( x ) = ϕ i ( x ) − G 0( M ) ( x ) ∗ M i ( x ) + ϕ N i ( x )( x ) = A ( x ) : ε i ( x ) + B ( x ) : ϕ i ( x ) M i ( x ) = B ( x ) : ε i ( x ) + D ( x ) : ϕ i ( x )

(10)

And for FoPT it is presented as:

  

ε i + 1 ( x ) = ε i ( x ) − G 0( NN ) ( x ) ∗ N i ( x ) + ε ϕ i + 1 ( x ) = ϕ i ( x ) − G 0( MM ) ( x ) ∗ M i ( x ) −G 0( MQ ) ( x ) ∗ Q i ( x ) + ϕ γ i + 1 ( x ) = γ i ( x ) − G 0( QQ ) ( x ) ∗ Q i ( x ) −G 0( QM ) ( x ) ∗ M i ( x ) + γ

(11)

N i ( x ) = A ( x ) : ε i ( x ) + B ( x ) : ϕ i ( x ) M i ( x ) = B ( x ) : ε i ( x ) + D ( x ) : ϕ i ( x ) Q i ( x ) = S ( x ) : γ i ( x )

 

In what follows, an enhanced discrete algorithm for FoPT is presented, and a similar approach can be also applied to (CPT) only with the shearing contributions excluded from consideration.

Algorithm 1: Solving the cell problem in First-order plate theory Data: A ( x ) , B ( x ) , D ( x ) , S ( x ) , ε , ϕ , γ , e f begin A 0 , B 0 , D 0 , S 0 and  G 0( NN ) , G 0( QQ ) , G 0( QM ) , G 0( MQ ) ,  G 0( MM ) ε 0 ( x ) ←− ¯ ε , ϕ 0 ( x ) ←− ¯ ϕ , γ 0 ( x ) ←− ¯ γ N 0 ( x ) ←− A ( x ) : ε 0 ( x ) + B ( x ) : ϕ 0 ( x ), M 0 ( x ) ←− D ( x ) : ϕ 0 ( x ) + B ( x ) : ε 0 ( x ), Q 0 ( x ) ←− S ( x ) : γ 0 ( x ) e c ←− + ∞ while e c > e f do Iteration: i ←− i + 1 Fourier Transformation of the stress and moment resultants:  N i ( ξ ) = FFT  N i ( x )  ,  M i ( ξ ) = FFT  M i ( x )  ,  Q i ( ξ ) = FFT  Q i ( x )  Calculate the residual: e c = Res   N i ( ξ ) ,  M i ( ξ ) ,  Q i ( ξ )  Calculate the new strain and curvature distribution:  ε i + 1 ( ξ ) =  ε i ( ξ ) −  G 0( NN ) ( ξ ) :  N i ( ξ ), and  ε i + 1 ( 0 ) = ε  ϕ i + 1 ( ξ ) =  ϕ i ( ξ ) −  G 0( MM ) ( ξ ) :  M i ( ξ ) −  G 0( MQ ) ( ξ ) :  Q i ( ξ ), and  ϕ i + 1 ( 0 ) = ϕ  γ i + 1 ( ξ ) =  γ i ( ξ ) −  G 0( QQ ) :  Q i ( ξ ) −  G 0( QM ) ( ξ ) :  M i ( ξ ), and  γ i + 1 ( 0 ) = ¯ γ Inverse Fourier Transformation of the strain and curvature vectors: ε i + 1 ( x ) = JFFT   ε i + 1 ( ξ )  , ϕ i + 1 ( x ) = JFFT  ϕ i + 1 ( ξ )  , γ i + 1 ( x ) = JFFT   γ i + 1 ( ξ )  Calculate the new stress and moment resultants:

N i + 1 ( x ) = A ( x ) : ε i + 1 ( x ) + B ( x ) : ϕ i + 1 ( x ) M i + 1 ( x ) = D ( x ) : ϕ i + 1 ( x ) + B ( x ) : ε i + 1 ( x ) Q i + 1 ( x ) = S ( x ) : γ i + 1 ( x ) Result: ε f ( x ), ϕ f ( x ), γ f ( x ), N f ( x ), M f ( x ), Q f ( x )