  For more than 100 years, chemical, physical, and material scientists have
proposed competing constitutive models to best characterize the behavior of
natural and man-made materials in response to mechanical loading. Now, computer
science offers a universal solution: Neural Networks. Neural Networks are
powerful function approximators that can learn constitutive relations from
large data without any knowledge of the underlying physics. However, classical
Neural Networks ignore a century of research in constitutive modeling, violate
thermodynamic considerations, and fail to predict the behavior outside the
training regime. Here we design a new family of Constitutive Artificial Neural
Networks that inherently satisfy common kinematic, thermodynamic, and physic
constraints and, at the same time, constrain the design space of admissible
functions to create robust approximators, even in the presence of sparse data.
We revisit the non-linear field theories of mechanics and reverse-engineer the
network input to account for material objectivity, symmetry, and
incompressibility; the network output to enforce thermodynamic consistency; the
activation functions to implement physically reasonable restrictions; and the
network architecture to ensure polyconvexity. We demonstrate that this new
class of models is a generalization of the classical neo Hooke, Blatz Ko,
Mooney Rivlin, Yeoh, and Demiray models and that the network weights have a
clear physical interpretation. When trained with classical benchmark data for
rubber, our network autonomously selects the best constitutive model and learns
its parameters. Our findings suggests that Constitutive Artificial Neural
Networks have the potential to induce a paradigm shift in constitutive
modeling, from user-defined model selection to automated model discovery. Our
source code, data, and examples are available at
https://github.com/LivingMatterLab/CANN.
