  In this paper, a new data-driven multiscale material modeling method, which
we refer to as deep material network, is developed based on mechanistic
homogenization theory of representative volume element (RVE) and advanced
machine learning techniques. We propose to use a collection of connected
mechanistic building blocks with analytical homogenization solutions which
avoids the loss of essential physics in generic neural networks, and this
concept is demonstrated for 2-dimensional RVE problems and network depth up to
7. Based on linear elastic RVE data from offline direct numerical simulations,
the material network can be effectively trained using stochastic gradient
descent with backpropagation algorithm, enhanced by model compression methods.
Importantly, the trained network is valid for any local material laws without
the need for additional calibration or micromechanics assumption. Its
extrapolations to unknown material and loading spaces for a wide range of
problems are validated through numerical experiments, including linear
elasticity with high contrast of phase properties, nonlinear history-dependent
plasticity and finite-strain hyperelasticity under large deformations.
  By discovering a proper topological representation of RVE with fewer degrees
of freedom, this intelligent material model is believed to open new
possibilities of high-fidelity efficient concurrent simulations for a
large-scale heterogeneous structure. It also provides a mechanistic
understanding of structure-property relations across material length scales and
enables the development of parameterized microstructural database for material
design and manufacturing.
