Discrete non-local interactions formulation of deep material networks for damage-enhanced constitutive material models

The Deep Material Network (DMN) has emerged as an efficient surrogate model for substituting direct nonlinear finite element analyses of heterogeneous material volume elements in the context of multi-scale simulations. Its extrapolation capabilities in terms of both the loading scenarii and the phases constitutive behaviours, as well as its thermodynamic consistency, broaden its applicability as compared to more “black-box” data-driven approaches such as neural networks. However, existing DMN applications do not handle phases modelled with continuum damage laws in a robust way.
A DMN has a tree architecture of nodes, in which basic material phases are represented by the leaf or material nodes located at the bottom of the tree. The effective or homogenised heterogeneous response is obtained at the root node through a sequence of interaction operations performed across the intermediate layers. When damage variables are introduced at the leaf nodes, damage tends to localise within a single node, leading to numerical instabilities. Moreover, the influence of this localised damage diminishes as it propagates through the network, resulting in an unreliable homogenised response.
To address this limitation—analogous to the challenge faced in finite element analyses with local damage models and the associated loss of solution uniqueness—we propose a discrete non-local damage formulation within the DMN framework. This approach enhances the numerical stability when damage initiates at the leaf nodes and enables the network to predict the homogenised response during the softening stage in a more robust and reliable way.