Data‐Driven Stress Prediction for Thermoplastic Materials

The present study applies two different machine learning (ML) algorithms to predict the stress‐strain mapping for the non‐linear behaviour of thermoplastic materials: a Long Short‐Term Memory (LSTM) algorithm and a Feed‐Forward Neural Network (FFNN). The approach of this work requires the generation of the stress‐strain curve for specific material parameters. The training data are obtained from the von Mises material law and the Ramberg‐Osgood equation. The four combinations of ML algorithms with constitutive laws are evaluated and show a good agreement with numerical data.


Introduction
Many mathematical models are available to describe the physical behavior of new materials such as thermoplastics. In the material modeling of these new materials, a discrepancy between simulation results and real physical behavior is still observed in many cases. Artificial neural networks can help bridge this gap. Machine learning algorithms are well suited to improve constitutive models by directly incorporating new numerical or even experimental data. To achieve this, an artificial neural network is trained on data created from two different material laws: a von Mises material with isotropic hardening and the Ramberg- The mathematical description of the elastic-plastic flow is based on the following three rules: a) Hooke's law (eq. 1), where the elastic response is described as a linear relationship between strain ε and stress σ which depends on the material parameters Young's modulus E and Poisson's ratio ν, b) yield criterion ϕ (eq. 2), which states that in the elastic response the yield stress σ 0.2 , considered to be the stress at which the material has a plastic deformation of 0.2%, cannot be exceeded by the applied stress σ VM , otherwise plastification begins, and c) hardening rule (eq. 3) that describes the effect of plastic deformation on the yield condition. Isotropic hardening is sufficient for plastic deformation under monotonic loading conditions. The hardening modulus κ can be specified by the slope of the uniaxial strain-stress response, in this case, a constant value.
With regard to the material parameters of the von Mises model C = [σ 0.2 , κ, ν, E], random values in a specific range are generated:

Ramberg-Osgood equation
The Ramberg-Osgood [3] equation is presented in eq. 4. Eq. 5 shows the Hill modification [4], which models the exponent n in terms of the material parameters σ 0.2 and σ 0.01 The following random values for the defined material parameters C = [σ 0.2 , σ 0.01 , E] in the Ramberg-Osgood model are generated in the specified ranges:

Machine learning algorithms
The problem at hand is a regression task. For the training, 500 different combinations of material parameters with 100 curve points are generated. The distribution of the dataset is 60% for the training, 20% for testing, and 20% for the validation. In both cases, the training is performed with the Adam optimizer [5]. The network consists of input and output layers as well as three fully connected hidden layers. The objective of the FFNN is, given specific material parameters C and one strain value of the curve ε i , to predict the stress at this respective strain value σ i = f (ε i , C).
• Long Short-Term Memory (LSTM): Six hidden layers are used, of which three are LSTMs and the last three are fully-connected layers. The LSTM receives as input five consecutive pairs of strains and stresses [(ε i , σ i ), ..., (ε i+4 , σ i+4 )] on the curve, along with the material parameters C. It returns the predicted stress σ i+5 at the specific strain value ε i+5 . Fig. 4 show two examples of predicted curves by the FFNN for data generated with the von Mises model on the left and data generated by the Ramberg-Osgood equation on the right, both of which are from the test set. Each curve progression corresponds to one set of material parameters. The ground truth is marked with a solid line and 20 predicted data points of the test set are symbolised by different shaped points. There is a strong overlap between actual and predicted values. The R 2 -score of R 2 > 0.999 for the two FFNNs supports the reliability of the results. Fig. 5 and Fig. 6 show a stress-strain curve of the von Mises and Ramberg-Osgood models with the LSTM. The LSTM reaches an R 2 -score of R 2 > 0.797 and R 2 > 0.836 for von Mises and Ramberg-Osgood respectively and does not perfectly match the real curve but follows the general tendency. This is likely due to the hyperparameters of the model not being finetuned for the problem. Future work consists of this hyperparameter optimization and finding the optimal number of stress-strain pairs to feed into the LSTM to predict the curve development.