A hybrid FEM-NN optimization method to learn the physics-constrained constitutive relations from full-field data

Neural networks (NNs) have demonstrated strong capabilities of representing high-dimensional, complex functional relations, and hence have been widely used to characterize complex constitutive relations for various types of materials, such as polycrystals, polymers, etc. However, to construct a reliable NN-based constitutive model, a considerable amount of data, i.e. stress-strain states along different loading paths is needed, which can be expensive to collect. To address such challenge, we develop a hybrid finite element method (FEM) - NN optimization framework to learn complex hyperelastic constitutive relations from full-field data. The key advantage of this framework is that it can make use of the non-uniform displacement field due to the geometric inhomogeneities for training NN-based constitutive models. Since such data can provide many different stress-strain states in a single test, it can greatly reduce the number of experiments needed for the training of NNs. Besides, we adopt a mechanics-informed neural network (MINN) as our architecture to ensure that our NN-based models satisfy all necessary physical constraints by construction, such as objectivity, material symmetry, polyconvexity, etc. Such architecture is also key to the convergence of our optimization framework. We then use both synthetic and experimental data to test the performance of our proposed framework on various isotropic hyperelastic materials. Results show that our optimization framework can be used to train NN-based constitutive models for hyperelastic materials with high accuracy and efficiency using data generated from simple tests, which can also be easily adapted to characterize complex constitutive models for a broader range of materials.