A data-driven multiscale scheme for anisotropic finite strain magneto-elasticity

TL;DR

The paper addresses modeling anisotropic finite-strain magneto-elastic behavior of structured MREs by coupling microscale homogenization with a macroscale physics-augmented neural network (PANN). It deploys a decoupled data-driven multiscale scheme: microscale RVEs under varied $\boldsymbol{F}$ and $\boldsymbol{B}$ generate homogenized data that trains a macroscale PANN to predict $\bar{\boldsymbol{P}}^{tot}$ and $\bar{\boldsymbol{H}}$, while enforcing thermodynamic consistency, objectivity, and symmetry. The approach extends PANNs to transverse isotropy, includes a two-step training procedure with direction learning, and demonstrates magnetostriction in a spherical MRE, validating macro predictions against microscale simulations. This framework enables efficient, physically constrained macroscopic simulations for MREs and can be extended to more complex microstructures and dissipative effects, providing a scalable tool for design and optimization of magneto-elastic composites.

Abstract

In this work, we develop a neural network-based, data-driven, decoupled multiscale scheme for the modeling of structured magnetically soft magnetorheological elastomers (MREs). On the microscale, sampled magneto-mechanical loading paths are imposed on a representative volume element containing spherical particles and an elastomer matrix, and the resulting boundary value problem is solved using a mixed finite element formulation. The computed microscale responses are homogenized to construct a database for the training and testing of a macroscopic physics-augmented neural network model. The proposed model automatically detects the material's preferred direction during training and enforces key physical principles, including objectivity, material symmetry, thermodynamic consistency, and the normalization of free energy, stress, and magnetization. Within the range of the training data, the model enables accurate predictions of magnetization, mechanical stress, and total stress. For larger magnetic fields, the model yields plausible results. Finally, we apply the model to investigate the magnetostrictive behavior of a macroscopic spherical MRE sample, which exhibits contraction along the magnetic field direction when aligned with the material's preferred direction.

Figures (16)

• Figure 1: Decoupled data-driven multiscale scheme: Macroscopic magneto-mechanical loads are applied to the RVE, and the resulting microscale fields are computed with the FEM from the known material behavior of the constituents. Homogenization yields a macroscale dataset for training the PANN model, which allows efficient macroscale FE simulations without explicitly solving microscopic BVPs.
• Figure 2: Illustration of the macroscopic PANN model for transversely isotropic magneto-elasticity. The input and output of the model are displayed in blue, trainable variables in green, the neural network with its input and output normalization in yellow and all remaining variables in red. The layout was inspired by Linden23Kalina24.
• Figure 3: Sampled and filtered mechanical invariants in the sampling ranges given in Table \ref{['tab:sampling_params']} are shown in blue. The load cases for which convergence could be achieved during the FE microscale simulation are shown in orange and are used for the training and testing of the PANN model.
• Figure 4: Sampled and filtered coupled magneto-mechanical invariants in their SI base units in the sampling ranges given in Table \ref{['tab:sampling_params']} are shown in blue. The load cases for which convergence could be achieved during the FE microscale simulation are shown in orange and are used for the training and testing of the PANN model.
• Figure 5: Section of the RVE, periodically extended in $z$-direction to visualize the particles chains. The cross-section of the complete RVE is shown in Figure \ref{['fig:microscale_fields']}.

Theorems & Definitions (2)

• Remark 1
• Remark 2