On the micromechanics of deep material networks

Abstract We investigate deep material networks (DMNs), recently introduced by Liu-Wu-Koishi [Comput. Method Appl. M., vol. 345, pp. 1138–1168, 2019], from the viewpoint of classical micromechanics at small strains. We aim to establish the basic micromechanical principles of deep material networks, shed light on the characteristics of the building blocks and introduce a simple, robust and fast solution technique for inelastic deep material networks. In their original formulation, DMNs are solely trained by linear elastic data, but applied to nonlinear and inelastic problems with astonishing accuracy. We clarify this phenomenon theoretically by showing that, to first order in the strain rate, the effective inelastic behavior of composite materials is determined by linear elastic localization. Our argumentation applies to arbitrary microstructures comprising nonlinear generalized standard materials at small strains. The main technical tool is a Volterra series approximation of the stress of a generalized standard material, which we adapt from nonlinear dynamical systems theory. Next, we establish that deep material networks inherit thermodynamical consistency and stress-strain monotonicity from their phases. This property roots in the definition of the DMN as a tree of hierarchical laminates and contrasts with other applications of neural networks to the approximation of material laws, where consistency and monotonicity typically cannot be guaranteed far away from the training set. Last but not least, we introduce rotation-free DMNs with arbitrary directions of lamination and exploit a novel formulation, uniting the implementation of DMNs of arbitrary tree topology and multi-phase laminates, and apply our insights to microstructures of industrial complexity.