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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ration of the fast Fourier transformation (FFT) Muller (1998). Consequently, this method is often referred to as the FFT-based homogenisation. The pioneering FFT-based homogenisation approach facilitated simulation on solid composite continua, abiding by the linear elasticity governing equation, and revealed the explicit Green’s function for isotropic reference media Li et al. (2022, 2023). In addition to the solid formulation-based cell problems, which have seen extensive development, researchers have also incorporated plate formulation in addressing cell problems. Plate models e ff ectively economizes on computational resources compared to solid formulation, as it ignores out-of-plane behaviour of plate structures in thin plate models which takes minimal e ff ect on the overall properties of plate structures Reddy (2006, 2003). Several studies have also demonstrated the feasibility of employing plate theories in homogenisation problems using plate models to solve cell problems Dong et al. (2019); Helfen and Diebels (2013); Bleyer and De Buhan (2014); Muller, Klarmann, & Gruttmann (2022). Nonetheless, these studies resorted to traditional numerical methods like FEM for plate cell problems, which have proven to be computationally expensive compared to the FFT-based approach. Hence, the development of an FFT-based solver for plate-model-based cell problems holds substantial promise. In light of this, the present work introduces an FFT-based approach for thin plate structures, wherein periodic Lippmann-Schwinger Equations are introduced for the governing equations of plate models. The solutions of the Lippmann-Schwinger equations for isotropic reference materials are deduced and articulated via explicit Green’s function. The remainder of this paper is structured as follows: Section 2 unveils the proposed Lippmann-Schwinger Equa tions for the Classical and First-order plate models, along with the derived Green’s function serving as the solution. Section 3 discusses the algorithmic implementation of the proposed FFT-based method. Section 4 o ff ers two case stud ies involving di ff erent structures employing the developed method. Finally, Section 5 provides concluding remarks.

2. Methodology

Among the variety of plate theories, the Classical Plate Theory (CPT) and the First-Order Plate Theory (FoPT) are most prevalently employed. In general practice, the Classical Plate Theory (CPT) finds application in plates with a thickness to width or length ratio of less than 10%, whereas the First-order Plate Theory (FoPT) is employed to handle plates with a ratio exceeding 10% . A brief overview of the two theories is given in this section, following with the theoretical introduction of the FFT-based solvers for each of them.

2.1. Classical Plate Theory

The governing equation of the Classical Plate Theory is expressed as Reddy (2006, 2003):

  

N = A ε + B ϕ M = D ϕ + B ε ∇· N = 0 ∇·∇ M = 0

(1)

where N and M represents the stress and moment resultants, respectively, ∇ is the nabla operator. A , B and D denote the ABD constitutive matrices in plate models. ε and ϕ represent the strain and curvature. In FFT-based homogeni sation approach, the above problem can be considered as a preliminary problem of a reference linear elastic material subjected to a polarisation field τ :    N = A 0 ε + B 0 ϕ + τ ( N ) M = D 0 ϕ + B 0 ε + τ ( M ) ∇· N = 0 ∇·∇ M = 0 (2)