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

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where t M is the traction vector at the macroscale, σ m the stress field at the microscale, and n the orientation of the adhesive layer at the microscale. A summary of the scale transition relations and the principle of scale separation between the homogeneous macroscale and the heterogeneous microscale is presented in Figure 1b. For the reasons already mentioned in the beginning of Section 2, it is numerically advantageous to apply only monotonic load paths in FFT-based homogenization with our in-house solver. Thus, we propose a method for training data generation, which is based on the e ff ective stress concept (Lemaitre (1996), pp. 12-14) and does not require unloading paths to estimate D M and u pl . Owing to the lack of data, it cannot be ensured that the unloading curves of the trained model correspond to the ones from FFT-based homogenization. It should also be mentioned that only monotonic load paths are usually used in the experimental parameter identification as well and further investigations on the unloading behavior are out of the scope of this work. Three simulations per volume fraction with monotonically increasing u are thereby needed: In a first simulation, the linear elastic constitutive laws are used only to calculate the sti ff ness K ( v f ) of adhesive layer. In a second simulation, an elastic-plastic constitutive law without damage is assumed for the polymer in order to estimate the macroscopic, undamaged e ff ective stress and the plastic separation by ˆ σ u , v f = K v f u − u pl u , v f ⇒ u pl u , v f = u − ˆ σ u , v f K v f . (7) In the last simulation, the elastic-plastic constitutive law including the damage model, as previously described, is used for the polymer. From this simulation, the macroscopic damage in the adhesive layer is estimated via σ u , v f = 1 − D M u , v f ˆ σ u , v f ⇒ D M u , v f = 1 − . (8)

σ u , v f ˆ σ u , v f

The values of plastic separation and damage at each volume fraction and separation were then used as training data for the artificial neural network.

2.3. Training of the neural networks

The FFNNs with 5 hidden layers and 50 neurons in each layer were created by means of the Python package PyTorch (Paszke et al. (2019)) using ReLU (Rectified Linear Unit) activation functions. The training was performed with the Adam optimizer and a learning rate of 10 − 5 , minimizing the Mean Squared Error (MSE) loss function. All other parameters for the Adam optimizer were the default ones in the PyTorch package. Furthermore, a batch size of 20 was used during the 2000 training epochs and the data set was randomly split into 10 % validation data and 90% training data. The training and the validation loss for the damage FFNN was 2 . 89 · 10 − 5 and 9 . 19 · 10 − 5 , respectively, whereas the training and the validation loss was 1 . 94 · 10 − 9 and 1 . 73 · 10 − 9 for the one predicting the plastic separation. The model in Equation 1 including the artificial neural networks was implemented as UMAT subroutine in Abaqus, whereby the tangent sti ff ness is calculated numerically by a forward di ff erence approximation.

3. Results and discussion

The resulting TSLs are shown in Figure 2a. It can be observed that with an increase of the volume fraction, there is a slight increase in both sti ff ness and strength. Also, with an increase of the volume fraction, a decrease of the critical energy release rate (ERR), i.e., the integral of the TSL (Equation 9), occurs. This coincides with the findings of Santos et al. (2022), who experimentally investigated the e ff ect of hollow glass spheres on the fracture behavior and obtained similar results for sti ff ness and critical ERR. Furthermore, the final damage field is also depicted in Figure 1a. Here