Equilibrium-based convolution neural networks for constitutive modeling of hyperelastic materials

Recently, various machine learning methods have been developed to model the stress-strain constitutive behavior of materials. In these methods, a large database of multi-axial stress versus strain curves is indispensable for training and cross-validation. However, it is usually formidable to experimentally obtain such curves by noting that, although strain can be easily measured, stress can only be extracted for limited specimens with a simple and uniform stress field upon loading. Moreover, most developed approaches are based upon supervised learning and ignore known physical laws, resulting in limited interpretability and generalization. In this paper, an equilibrium-based convolution neural network (ECNN) is proposed to model the constitutive behavior of hyperelastic materials. Only strain and externally applied force are taken as input for training while the hard-to-measure stress is considered as internal variables. Therefore, a large database can be easily generated with a single non-uniformly deformed specimen with spatially dependent strain. The equation of equilibrium is embedded in the architecture of the ECNN as a constraint to implicitly establish the correspondence between the internal variables and the stress components and, consequently, the constitutive stress-strain relationship. Moreover, the ECNN is unsupervised by noting that training is not against the output stress. Numerical and experimental examples are presented to show that the developed ECNN can faithfully extract the stress corresponding to local strain, model the multiaxial stress-strain constitutive behavior of hyperelastic materials, and surrogate equation-based conventional constitutive models of hyperelastic materials.