# Exploring the 3D architectures of deep material network in data-driven multiscale mechanics

arXiv: https://arxiv.org/abs/1901.04832 · HTML: https://mlsanity.com/paper/1901.04832/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jan 2019.

## Built on by (most cited first, 14 of 14)

- [A review on data-driven constitutive laws for solids](https://mlsanity.com/paper/2405.03658/) (2024)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Multiscale modeling of inelastic materials with Thermodynamics-based Artificial Neural Networks (TANN)](https://mlsanity.com/paper/2108.13137/) (2021)
- [Local approximate Gaussian process regression for data-driven constitutive laws: Development and comparison with neural networks](https://mlsanity.com/paper/2105.04554/) (2021)
- [Lossless Multi-Scale Constitutive Elastic Relations with Artificial Intelligence](https://mlsanity.com/paper/2108.02837/) (2021)
- [Deep autoencoders for physics-constrained data-driven nonlinear materials modeling](https://mlsanity.com/paper/2209.04416/) (2022)
- [Stress representations for tensor basis neural networks: alternative formulations to Finger-Rivlin-Ericksen](https://mlsanity.com/paper/2308.11080/) (2023)
- [Model-data-driven constitutive responses: application to a multiscale computational framework](https://mlsanity.com/paper/2104.02650/) (2021)
- [LS-DYNA Machine Learning-based Multiscale Method for Nonlinear Modeling of Short Fiber-Reinforced Composites](https://mlsanity.com/paper/2301.02738/) (2023)
- [Multi-Scale Simulation of Complex Systems: A Perspective of Integrating Knowledge and Data](https://mlsanity.com/paper/2306.10275/) (2023)
- [Orientation-aware interaction-based deep material network in polycrystalline materials modeling](https://mlsanity.com/paper/2502.02457/) (2025)
- [Cell division in deep material networks applied to multiscale strain localization modeling](https://mlsanity.com/paper/2101.07226/) (2021)
- [Systematic Performance Assessment of Deep Material Networks for Multiscale Material Modeling](https://mlsanity.com/paper/2602.07192/) (2026)
- [Deep material network for homogenization of piezoelectric composites](https://mlsanity.com/paper/2606.22566/) (2026)

## Abstract (arXiv)

This paper extends the deep material network (DMN) proposed by Liu et al. (2019) to tackle general 3-dimensional (3D) problems with arbitrary material and geometric nonlinearities. It discovers a new way of describing multiscale heterogeneous materials by a multi-layer network structure and mechanistic building blocks. The data-driven framework of DMN is discussed in detail about the offline training and online extrapolation stages. Analytical solutions of the 3D building block with a two-layer structure in both small- and finite-strain formulations are derived based on interfacial equilibrium conditions and kinematic constraints. With linear elastic data generated by direct numerical simulations on a representative volume element (RVE), the network can be effectively trained in the offline stage using stochastic gradient descent and advanced model compression algorithms. Efficiency and accuracy of DMN on addressing the long-standing 3D RVE challenges with complex morphologies and material laws are validated through numerical experiments, including 1) hyperelastic particle-reinforced rubber composite with Mullins effect; 2) polycrystalline materials with rate-dependent crystal plasticity; 3) carbon fiber reinforced polymer (CFRP) composites with fiber anisotropic elasticity and matrix plasticity. In particular, we demonstrate a three-scale homogenization procedure of CFRP system by concatenating the microscale and mesoscale material networks. The complete learning and extrapolation procedures of DMN establish a reliable data-driven framework for multiscale material modeling and design.