PSI - Issue 52

Pascal Alexander Happ et al. / Procedia Structural Integrity 52 (2024) 401–409 Author name / Structural Integrity Procedia 00 (2019) 000–000

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can have highly intricate shapes (see Park et al. (2007); Seo et al. (2006); Niu et al. (2009) and Fig. 1 below).

Fig. 1: Some particle shapes encountered in this research.

Micromechanical methods for mechanical properties estimation such as those developed by Reuss (1929); Voigt (1889); Benveniste (1987); Hill (1963) as well as the computational methods like finite element method (FEM) applied in works proposed by Trofimov et al. (2017); Zhang et al. (2014); Fritzen and Böhlke (2011); Gitman et al. (2007); Gusev (1997); Happ and Piat (2022) can be used to predict the material behavior of particles reinforced composites, with the aim to reduce their development time. But these methods have limitations, where either the theory restricts the approximation shape of the studied particle, such as only using ellipsoids, or the complexity of the shape can be prohibitively costly for the FE method to compute. Complex particle shapes such as those presented in this work (see Fig. 1) can be too computationally demanding to directly use in the FEM analysis, especially when considering multiple particles. The question dealt with in the present research is: How can complex particle shapes be approximated to reduce the computational cost while still being able to account for the contributions of the undulated particle shape onto the overall elastic properties of the composite? This work proposes a method by employing the use of surrogates as suggested by Gentile et al. (2018); Aggarwal et al. (2016); Asher et al. (2015); Forrester and Keane (2009); Queipo et al. (2005) to approximate arbitrary particle shapes reducing the computational cost while still being able to evaluate the contribution of the shape onto the effective elastic properties of the composite. The remainder of the article is structured as follows. Section 2 introduces the equations that are used to describe the undulated surface of the studied particles. Section 3 introduces the method for creating the surrogate models. The analytic schemes proposed by Reuss and Voigt as well as the Particle Swarm Optimization (PSO) are stated and the algorithms used for the implementation of the new homogenization schemes based on surrogates are developed. The results of the numerical studies are shown and discussed in Section 4. A short conclusion is given in Section 5. 2. Numerical Modeling of the Microstructure 2.1 Particle creation using Laplace’s Spherical Harmonics The complex particle shapes displayed in Fig. 1 can be approximated using an analytic approach, where a specific undulation is added onto the corresponding spherical particle. The undulations are created by using the theory of Laplace’s Spherical Harmonics (LSH) as stated by Müller (1966) and Feuerbacher (2019), described in the following way: � � ( , ) = �2 2+ 1 ( − ( )! + )! � � (cos ) ��� , with � � (cos ) representing the associated Legendre-Polynomial and and denote degree and order respectively.