PSI - Issue 42

Pauline Herr et al. / Procedia Structural Integrity 42 (2022) 498–505 P. Herr et al. / Structural Integrity Procedia 00 (2019) 000–000 P. Herr et al. / Structural Integrity Procedia 00 (2019) 000–000

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Moreover, from an engineering point of view, it is advantageous to have estimates of the global (macroscopic) damage and plastic deformation for technical design purposes. The procedure of determining the full ML-based model consists of the following three steps: Generation of virtual RVEs for di ff erent filler volume fractions, data generation using the results from FFT-based homogenization and training of the neural networks. Moreover, from an engineering point of view, it is advantageous to have estimates of the global (macroscopic) damage and plastic deformation for technical design purposes. The procedure of determining the full ML-based model consists of the following three steps: Generation of virtual RVEs for di ff erent filler volume fractions, data generation using the results from FFT-based homogenization and training of the neural networks. In FFT-based homogenization, it can only be dealt with RVEs which are discretized with a uniform grid. Therefore, voxelized RVEs with a size of 0.8 mm x 0.8 mm x 0.3 mm and a voxel size of 5 µ m were generated for each filler volume fraction investigated using Random sequential adsorption (RSA). Owing to the cohesive zone model, the entire thickness of the adhesive layer is represented by the RVE. Besides the glass beads of 35 µ m diameter, pores were also added to the polymer matrix, as they are usually induced during the manufacturing process of adhesive bonds. The volume fraction of the pores of 42 µ m diameter was at about 4.2% in each RVE, which is both in the same order of magnitude as found in Dumont et al. (2020) by computer tomography investigations of industrial adhesives. In the RSA algorithm, glass beads and pores are randomly distributed within the adhesive layer one after another until the specified volume fraction is reached, whereby overlap of particles and pores was avoided. Furthermore, small sti ff layers were added in the orientation direction of the cohesive zone, in our case the z direction (without loss of generality of the method). The reason for this is the intrinsic periodicity of FFT-based homogenization, which is not valid in this direction owing to the adjacent bonded parts. If the layers are significantly sti ff er than the structure itself, they approximately eliminate these intrinsic periodic boundary conditions, see the FFT-based homogenization scheme for cohesive zones in Bo¨deker et al. (2022) for more details. An exemplary RVE resulting from the RSA is shown in Figure 1a. Volume fractions of glass beads of 1.67%, 5%, 10% and 20% were studied in this work. a In FFT-based homogenization, it can only be dealt with RVEs which are discretized with a uniform grid. Therefore, voxelized RVEs with a size of 0.8 mm x 0.8 mm x 0.3 mm and a voxel size of 5 µ m were generated for each filler volume fraction investigated using Random sequential adsorption (RSA). Owing to the cohesive zone model, the entire thickness of the adhesive layer is represented by the RVE. Besides the glass beads of 35 µ m diameter, pores were also added to the polymer matrix, as they are usually induced during the manufacturing process of adhesive bonds. The volume fraction of the pores of 42 µ m diameter was at about 4.2% in each RVE, which is both in the same order of magnitude as found in Dumont et al. (2020) by computer tomography investigations of industrial adhesives. In the RSA algorithm, glass beads and pores are randomly distributed within the adhesive layer one after another until the specified volume fraction is reached, whereby overlap of particles and pores was avoided. Furthermore, small sti ff layers were added in the orientation direction of the cohesive zone, in our case the z direction (without loss of generality of the method). The reason for this is the intrinsic periodicity of FFT-based homogenization, which is not valid in this direction owing to the adjacent bonded parts. If the layers are significantly sti ff er than the structure itself, they approximately eliminate these intrinsic periodic boundary conditions, see the FFT-based homogenization scheme for cohesive zones in Bo¨deker et al. (2022) for more details. An exemplary RVE resulting from the RSA is shown in Figure 1a. Volume fractions of glass beads of 1.67%, 5%, 10% and 20% were studied in this work. a b 2.1. Generation of virtual RVEs 2.1. Generation of virtual RVEs

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t M = σ m x · n

m = (0 0 u / t 0 0 0) T

E V

z

z

y

y

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Fig. 1. (a) RVE with damage field; (b) Scheme of scale transition. Fig. 1. (a) RVE with damage field; (b) Scheme of scale transition.

The (solid) glass beads were modeled as isotropic, linear elastic. The assumption was made here that their fracture behavior is negligible, and thus no damage model was applied. Moreover, a significant amount of plastic deformation of the adhesive layer can often be observed in experiments, e.g., Loh and Marzi (2019). Hence, the plastic and fracture behavior of the RVE are purely caused by the polymer in our model. For simplicity, we used a linear Drucker-Prager yield condition with linear hardening of the form Φ = σ vM − tan( β ) p − d 0 + H ¯ κ l = 0 (2) in the plasticity model to take into account the dependence of the hydrostatic pressure on the plastic behavior, which is typical for many polymers. σ vM , β and p are thereby the von Mises equivalent stress, friction angle of the model and the hydrostatic pressure, respectively. Additionally, d 0 and H represent the initial yield stress and the hardening modulus, whereas ¯ κ l is the (local) equivalent plastic strain. However, the linear Drucker-Prager is sometimes not su ffi cient to The (solid) glass beads were modeled as isotropic, linear elastic. The assu ption as ade here that their fracture behavior is negligible, and thus no damage model was applied. oreover, a significant a ount of plastic defor ation of the adhesive layer can often be observed in experiments, e.g., Loh and arzi (2019). ence, the plastic and fracture behavior of the RVE are purely caused by the polymer in our model. For simplicity, e used a linear rucker-Prager yield condition with linear hardening of the form Φ = σ vM − tan( β ) p − d 0 + H ¯ κ l = 0 (2) in the plasticity model to take into account the dependence of the hydrostatic pressure on the plastic behavior, which is typical for many polymers. σ vM , β and p are thereby the von Mises equivalent stress, friction angle of the model and the hydrostatic pressure, respectively. Additionally, d 0 and H represent the initial yield stress and the hardening modulus, whereas ¯ κ l is the (local) equivalent plastic strain. However, the linear Drucker-Prager is sometimes not su ffi cient to