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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The solution of this equation can be obtained by the periodic Lippman-Schwinger equation according to the periodicity assumed in a cell problem:  ε ( x ) = − G 0( NN ) ( x ) ∗ τ ( N ) ( x ) − G 0( NM ) ( x ) ∗ τ ( M ) ( x ) + ε ϕ ( x ) = − G 0( MM ) ( x ) ∗ τ ( M ) ( x ) − G 0( MN ) ( x ) ∗ τ ( N ) ( x ) + ϕ 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 ) (3) The Green’s function of the foregoing equation can be obtained in the Fourier space. Note here that all the variables in the Fourier space is denoted by the upper tilde ’  ’.  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  λ 0 ( m ) + 2 µ 0 ( m )  ξ i ξ j ξ k ξ h | ξ | 4  G 0( MN ) khi j = 0  G 0( MM ) khi j = − 1  λ 0 ( b ) + 2 µ 0 ( b )  ξ i ξ j ξ k ξ h | ξ | 4  G O ( NM ) khi j = 0 (4) for isotropic reference materials. The readers are referred to Li et al. (2023) for the derivation of the Green’s functions. The First-order Plate Theory (FoPT) follows the following governing equations since the introduction of the shear contribution Reddy (2006, 2003):    (5) where S denotes the shearing matrix of the constitutive model, γ denotes the shearing strain. Similarly, the auxiliary problem of FoPT can be defined as:    N = A 0 ε + B 0 ϕ + τ ( N ) M = D 0 ϕ + B 0 ε + τ ( M ) Q = S 0 γ + τ ( Q ) ∇· N = 0 ∇· M − Q = 0 ∇ Q = 0 (6)    2.2. First-order Plate Theory   N = A ε + B ϕ M = D ϕ + B ε Q = S γ ∇· N = 0 ∇· M − Q = 0 ∇ Q = 0

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