Viscoelasticty with physics-augmented neural networks: Model formulation and training methods without prescribed internal variables

We present an approach for the data-driven modeling of nonlinear viscoelastic materials at small strains which is based on physics-augmented neural networks (NNs) and requires only stress and strain paths for training. The model is built on the concept of generalized standard materials and is therefore thermodynamically consistent by construction. It consists of a free energy and a dissipation potential, which can be either expressed by the components of their tensor arguments or by a suitable set of invariants. The two potentials are described by fully/partially input convex neural networks. For training of the NN model by paths of stress and strain, an efficient and flexible training method based on a recurrent cell, particularly a long short-term memory cell, is developed to automatically generate the internal variable(s) during the training process. The proposed method is benchmarked and thoroughly compared with existing approaches. These include a method that obtains the internal variable by integrating the evolution equation over the entire sequence, while the other method uses an an auxiliary feedforward neural network for the internal variable(s). Databases for training are generated by using a conventional nonlinear viscoelastic reference model, where 3D and 2D plane strain data with either ideal or noisy stresses are generated. The coordinate-based and the invariant-based formulation are compared and the advantages of the latter are demonstrated. Afterwards, the invariant-based model is calibrated by applying the three training methods using ideal or noisy stress data. All methods yield good results, but differ in computation time and usability for large data sets. The presented training method based on a recurrent cell turns out to be particularly robust and widely applicable and thus represents a promising approach for the calibration of other types of models as well.