Theory and implementation of inelastic Constitutive Artificial Neural
  Networks
Hagen Holthusen Lukas Lamm Tim Brepols Stefanie Reese Ellen Kuhl
Nature has always been our inspiration in the research, design and
development of materials and has driven us to gain a deep understanding of the
mechanisms that characterize anisotropy and inelastic behavior. All this
knowledge has been accumulated in the principles of thermodynamics. Deduced
from these principles, the multiplicative decomposition combined with pseudo
potentials are powerful and universal concepts. Simultaneously, the tremendous
increase in computational performance enabled us to investigate and rethink our
history-dependent material models to make the most of our predictions. Today,
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, restrictions of energy and
pseudo potential, and consistent evolution guide us towards the architecture of
the iCANN satisfying thermodynamics per design. We combine feed-forward
networks of the free energy and pseudo potential with a recurrent neural
network approach to take time dependencies into account. We demonstrate that
the iCANN is capable of autonomously discovering models for artificially
generated data, the response of polymers for cyclic loading and the relaxation
behavior of muscle data. As the design of the network is not limited to
visco-elasticity, our vision is that the iCANN will reveal to us new ways to
find the various inelastic phenomena hidden in the data and to understand their
interaction. Our source code, data, and examples are available at
doi.org/10.5281/zenodo.10066805