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.

翻译：磁活性弹性体在机械载荷与磁场的共同作用下表现出大变形非线性响应，其有效行为强烈受微观结构非均匀性支配。这类材料的预测建模具有挑战性，因为其响应涉及强烈的磁-力耦合、大变形以及弹性体基质的近不可压缩特性。现有多尺度方法通常依赖交错策略或无法强鲁棒处理强耦合环境中近不可压缩性的公式。本文提出了一种完全耦合的计算均匀化框架，用于近不可压缩磁弹性复合材料，其中机械变形与静磁场在代表性体积单元（RVE）上以整体方式求解。微观尺度问题采用混合有限元离散化，包含拉格朗日位移自由度与基于Nédélec的磁矢势，从而实现对磁感应强度的旋度保持表示，以及对机械场和磁场的周期性边界约束。近不可压缩性通过J-bar稳定化处理，其中体积响应由单元平均膨胀控制，而等容响应则使用缩放变形梯度评估。本构行为基于可加自由能分解推导，包含超弹性、真空磁化及饱和型磁化贡献。该公式使得在大变形与强耦合条件下，对具有复杂颗粒分布的非均匀磁弹性复合材料进行稳健的三维RVE模拟成为可能。数值算例展示了颗粒相互作用、微观结构排列及夹杂物可压缩性如何影响变形模式与有效磁-力响应。