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Description
Mean-field homogenization theories mainly rely on the Eshelby problem, which provides solutions for eigenstrain concentration in an ellipsoidal inclusion embedded within an infinite matrix. In nature, however, microstructures often feature inclusions with non-ellipsoidal shapes for which no analytical Eshelby solutions exist [1]. To address this limitation, we aim to develop a generalizable solution that accommodates arbitrary inclusion shapes by adapting the recently developed deep material networks (DMNs).
DMNs [2, 3] learn the underlying microstructure of a material and are able to predict the material's linear and, to a certain extent, nonlinear macroscopic behavior. DMNs are hierarchical, tree-shaped artificial neural networks. Each neuron represents a classical laminate for which stiffness homogenization, strain concentration, and rotation operations are defined. They are typically trained on finite element (FE) results of geometrically exact representations of a specific microstructure configuration. The trained network can be used to predict the material's nonlinear behavior. However, DMNs are restricted to one single microstructure. To account for varying volume fractions of the material constituents, a varying orientation of inclusion-type constituents or a multi-scale hierarchical material is not straightforward and requires new training data generation and, thus, numerous additional FE simulations [1].
Inspired by these challenges, we propose a novel modeling framework utilizing DMNs to solve the Eshelby problem. Our approach, termed the Deep Eshelby Network (DEN), integrates seamlessly with classical mean-field homogenization theories, bypassing the need for exhaustive microstructure-specific training. Similar to the DMNs, the deep Eshelby network is a tree-shaped neural network where each neuron performs a stiffness homogenization and strain concentration based on the laminate homogenization theory. Unlike DMNs, which are trained based on homogenized stiffnesses, the DEN is trained on strain concentration tensors associated with the Eshelby problem, which are derived from FE simulations of superellipsoidal inclusions embedded in an infinite matrix. By leveraging superellipsoid parameterization, the DEN can predict strain concentration tensors for a wide range of physically occurring inclusion geometries. The proposed and already-trained DEN can be used to homogenize a large variety of multi-scale microstructures without the need for any additional training, overcoming the limitations of analytical approaches while being more efficient than FE simulations.
[1] Traxl Roland, and Lackner Roman (2018). Mech Mater, 126, 126–39.
[2] Liu Zeliang, and Wu C. T. (2019). J Mech Phys Solids, 127, 20–46.
[3] Gajek Sebastian, Schneider Matti, and Böhlke Thomas (2020). J Mech Phys Solids, 142, 103984.
