Theory and implementation of inelastic Constitutive Artificial Neural Networks

The two fundamental concepts of materials theory, pseudo potentials and the assumption of a multiplicative decomposition, allow a general description of inelastic material behavior. The increase in computer performance enabled us to thoroughly investigate the predictive capabilities of ever more complex choices for the potential and the Helmholtz free energy. Today, however, we have reached a point where materials and their models are becoming increasingly sophisticated. This raises the question: How do we find the best model that includes all inelastic effects to explain our complex data? Constitutive Artificial Neural Networks (CANN) may answer this question. Here, we extend the CANNs to inelastic materials (iCANN). Rigorous considerations of objectivity, rigid motion of the reference configuration, multiplicative decomposition and its inherent non-uniqueness, choice of appropriate stretch tensors, restrictions of energy and pseudo potential, and consistent inelastic evolution guide us towards the general architecture of the iCANN satisfying thermodynamics per design. We combine feed-forward networks of the Helmholtz free energy and pseudo potential with a recurrent neural network approach to take time dependencies into account. Specializing the general iCANN to visco-elasticity, we demonstrate that the iCANN is capable of autonomously discovering models for artificially generated data, the response of polymers at different stretch rates for cyclic loading as well as the relaxation behavior of muscle data. Since the design of the network is not limited to visco-elasticity, iCANNs might help to autonomously identify the inelastic phenomena of the material and subsequently select the most appropriate model. Here, our focus is on providing a thermodynamically consistent framework for inelastic material behaviors and how to incorporate this framework into neural networks in an architecture-based manner. Our source code, data, and examples are available at Holthusen et al. (2023a) ( https://doi.org/10.5281/zenodo.10066805).