PSI - Issue 72

Stefan Hildebrand et al. / Procedia Structural Integrity 72 (2025) 520–528

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for time step k+1, with the shear modulus G and overlined quantities being the deviatoric parts. Considering the deviatoric strain instead of total strains relieves the NN from discovering the isochoric character of the process by itself in order to optimize the accuracy of the results and the training effort as much as possible Bock et al. (2021). All inputs and outputs of the NN are scaled to the same order of magnitude, preferredly 100 which can be conducted by a MinMaxScaler Raju et al. (2020) or manually, based on the respective material. The data flow is illustrated in Fig. 1. The NN used in the following has 6 LeakyReLU layers with 64 neurons each and is trained for 85 epochs with batch size 8, 75 epochs Adam (lr = 0.00001) 10 epochs Adamax (lr = 0.000001) .

Fig. 1. Suggested NN schema.

3. Numerical validation A sensitivity analysis of the loss function contributions has shown that the purely data-driven approach exhibits insufficient stability for multiple load cycles unless all loss contributions described here are considered Hildebrand and Klinge (2024b). The surrogate training and validation data are generated by using the Armstrong-Frederick model implemented in RRM as suggested in Suchocki (2022). The applied material parameters correspond to steel alloy 4130 and are listed in Tab. 2 Rahman et al. (2008); Varelis (2010). They are E = 183 GPa, υ = 0.302, σ Y,0 =300MPa, 3 160GPa 2 C   and 3 510 2    . The training data include stress and strain histories generated by random Gaussian processes. Six independent time series are created, one for each independent component of the total strain tensor. This procedure avoids the introduction of biases in the data set and results in an equal amount of both positive and negative strain values which is necessary to account for arbitrary cyclic loading histories. Each strain history is chosen to contain 10 000 data points. The strain histories are multiplied by the factor ,0 5 Y E  , which leads to a sufficient coverage of the space with plastic yielding. The application of 256 strain histories is sufficient for to yield good accurate results. The chosen parameters for the data generation are listed in Tab. 1.

Table 1. Properties of data generation. Number of random walks

256 Gaussian random process generator Normal distribution Covariance kernel Exponentiated quadratic Meanvalue 0 Sampling interval [-25, 25] Samples per patch 1000 Number of patches 10 Scaling factor ,0 5 Y E 

3.1. Results

The test cases are set up for uniaxial deformation, pure shear, and random multiaxial strain history to give an insight in the accuracy in complex loading scanerios.