PSI - Issue 52

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

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4

whose solution can be reported as the periodic Lippman-Schwinger equation:

     ε ( x ) = − G 0( NN ) ( x ) ∗ τ ( N ) ( x ) + ε ϕ ( x ) = − G 0( MM ) ( x ) ∗ τ ( M ) ( x ) − G 0( MQ ) ( x ) ∗ τ ( Q ) ( x ) + ϕ γ ( x ) = − G 0( QQ ) ( x ) ∗ τ ( Q ) ( x ) − G 0( QM ) ( x ) ∗ τ ( M ) ( x ) + γ τ ( N ) ( x ) =  A ( x ) − A 0 ( x )  : ε ( x ) +  B ( x ) − B 0 ( x )  : ϕ ( x ) τ ( M ) ( x ) =  B ( x ) − B 0 ( x )  : ε ( x ) +  D ( x ) − D 0 ( x )  : ϕ ( x ) τ ( Q ) ( x ) =  S ( x ) − S 0 ( x )  : γ ( x )

(7)

in which the Green’s functions are stated as follows in the Fourier space. Note here that all the variables in the Fourier space is denoted by the upper tilde ’  ’.     ε kh = −  G 0( NN ) khi j  τ ( N ) i j  ϕ kh = −  G 0( MM ) khi j  τ ( M ) i j −  G 0( MQ ) khi  τ ( Q ) i  γ k = −  G 0( QQ ) ki  τ ( Q ) i −  G 0( QM ) ki j  τ ( M ) i j (8) where  G 0( NN ) khi j is a forth order tensor for the membrane solution,  G 0( QQ ) khi j ,  G 0( QM ) ki j ,  G 0( MQ ) khi and  G 0( MM ) ki are second, third, third and fourth order tensors, respectively, for the coupling solutions of the bending and shearing contributions. They are explicitly stated as:  G 0( NN ) khi j = 1 4 µ 0 ( m ) | ξ | 2  δ ki ξ h ξ j + δ hi ξ k ξ j + δ kj ξ h ξ i + δ hj ξ k ξ i  − λ 0 ( m ) + µ 0 ( m ) µ 0 ( m )  λ 0 ( m ) + 2 µ 0 ( m )  ξ i ξ j ξ k ξ h | ξ | 4  G 0( MM ) khi j = 1 2  µ 0 ( b ) | ξ | 2 + G 0   δ ih ξ j ξ k + δ ik ξ h ξ j  − ξ h ξ i ξ j ξ k | ξ | 2    1 µ 0 ( b ) | ξ | 2 + G 0 − 1 | ξ | 2  λ 0 ( b ) + 2 µ 0 ( b )      G O ( MQ ) khi =    δ ih ξ k + δ ik ξ h 2  µ 0 ( b ) | ξ | 2 + G 0  − ξ h ξ i ξ k | ξ | 2  µ 0 ( b ) | ξ | 2 + G 0     i  G 0( QM ) ki = δ ik µ 0 ( b ) | ξ | 2 + G 0 − ξ i ξ k | ξ | 2    1 µ 0 ( b ) | ξ | 2 + G 0 − 1 G 0     G 0( QQ ) ki j =    ξ i ξ j ξ k | ξ | 2  µ 0 ( b ) | ξ | 2 + G 0  − δ ik ξ j µ 0 ( b ) | ξ | 2 + G 0    i (9) Note here that the coupling terms between the in-plane and bending contributions  G 0( MN ) khi j and  G 0( NM ) khi j vanish since they are zero as indicated by Eq.4. The readers are referred to Li et al. (2023) for the derivation detail of the Green’s functions.

3. Implementation

Presented here is the algorithm dedicated to addressing a cell problem using CPT and FoPT . This algorithm is focusing on solving the periodic Lippman-Schwinger Equations, as expressed in Eq.3 and Eq.7, within the framework of a basic scheme Moulinec and Suquet (1998). The iterating scheme for CPT is presented as follows where i here