Magneto-active elastomers exhibit large, nonlinear deformations under combined mechanical loading and magnetic fields, and their effective behavior is strongly governed by microstructural heterogeneity. Predictive modeling of these materials is challenging because their response involves strong magneto-mechanical coupling, large deformations, and the nearly incompressible behavior of elastomeric matrices. Existing multiscale approaches often rely on staggered strategies or formulations that do not robustly treat near-incompressibility in strongly coupled settings. This work presents a fully coupled computational homogenization framework for nearly incompressible magnetoelastic composites in which the mechanical deformation and magnetostatic fields are solved monolithically on a representative volume element (RVE). The microscale problem uses a mixed finite-element discretization with Lagrangian displacement degrees of freedom and a N'ed'elec-based magnetic vector potential, enabling a curl-conforming representation of magnetic induction together with periodic boundary constraints for both mechanical and magnetic fields. Near-incompressibility is treated using J-bar stabilization, in which the volumetric response is controlled by the cell-averaged dilatation while the isochoric response is evaluated using a scaled deformation gradient. The constitutive behavior is derived from an additive free-energy decomposition with hyperelastic, vacuum magnetic, and saturation-type magnetization contributions. The resulting formulation enables robust three-dimensional RVE simulations of heterogeneous magneto-elastic composites with complex particle distributions under large deformations and strong coupling. Numerical examples show how particle interactions, microstructural arrangement, and inclusion compressibility influence deformation patterns and the effective magneto-mechanical response.

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