  The mechanical behavior of inelastic materials with microstructure is very
complex and hard to grasp with heuristic, empirical constitutive models. For
this purpose, multiscale, homogenization approaches are often used for
performing reliable, accurate predictions of the macroscopic mechanical
behavior of solids and structures. Nevertheless, the calculation cost of such
approaches is extremely high and prohibitive for real-scale applications
involving inelastic materials.
  Here, we propose the so-called Thermodynamics-based Artificial Neural
Networks (TANN) for the constitutive modeling of materials with inelastic and
complex microstructure. Our approach integrates thermodynamics-aware
dimensionality reduction techniques and thermodynamics-based deep neural
networks to identify, in an autonomous way, the constitutive laws and discover
the internal state variables of complex inelastic materials. The efficiency and
accuracy of TANN in predicting the average and local stress-strain response,
the free-energy and the dissipation rate is demonstrated for both regular and
perturbed two- and three-dimensional lattice microstructures in inelasticity.
TANN manage to identify the internal state variables that characterize the
inelastic deformation of the complex microstructural fields. These internal
state variables are then used to reconstruct the microdeformation fields of the
microstructure at a given state. Finally, a double-scale homogenization scheme
(FEMxTANN) is used to solve a large scale boundary value problem. The high
performance of the homogenized model using TANN is illustrated through detailed
comparisons with microstructural calculations at large scale. An excellent
agreement is shown for a variety of monotonous and cyclic stress-strain paths.
