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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The surrogate creation process is shown schematically in Fig. 3 and is carried out as the following: E.g. the particle with the undulation Y 6 3 is cut into a core and a number of layers (see Fig. 3a), the corresponding layered spherical particles are the surrogate model of the initial particle with the undulations (see Fig. 3b). The material-ratios of the particle and matrix material are then estimated in respect to each layer. Each shell layer of the surrogate has the exact same inner and outer radius as was used for the slicing of each layer of the accurate particle.

Fig. 3: Surrogate creation process. a) The initial particle is cut into layers; b) A layered surrogate model is created, where the material parameter of every layer is evaluated using the Algorithms 1-3. 3. Material modeling 3.1 Calculation of the effective elastic properties of the layers For calculation of the elastic properties of each layer, the equation that describes the elastic material behavior can be written as the following: = �� : , with , �� and are the stress tensor, the elastic tensor and the strain tensor respectively. The elementary solutions proposed by Reuss (1929) and Voigt (1889) for two phase composite were taken into consideration. The Reuss bound can be stated as the following: �� =h 1 C 1 +h 2 C 2 , with C 1 and C 2 being corresponding elastic tensors of the matrix and particles materials, h 1 and h 2 being corresponding the volume fraction of the matrix and particles of the composite. For the Voigt bound the following presentation can be made: �� =[h 1 (C 1 ) -1 +h 2 (C 2 ) -1 ] -1 . These two approximations were used for the calculation of the effective elastic properties of each layer in the surrogate model. This process is iterated starting from the core layer and carried through to the final outer layer. The layers are then stacked inside of each other starting with the spherical core, where the elastic parameters are the parameters of the particle material (see Fig. 3b). This process is described in more detail in Algorithm 1, Fig. 4. The calculation procedure for each layer (with exception of the starting layer) using the approximations by Voigt and Reuss are presented in Fig. 4, corresponding to Algorithm 2 and 3.