Development of a novel phenomenological–mechanistic model for composite magnetorheological foams under small-amplitude oscillatory shear

Magnetorheological (MR) foams are lightweight field-responsive functional materials whose dynamic stiffness and damping can be tuned by an external magnetic field. MR foams containing 40, 50, and 60 wt.% carbonyl iron particles (CIPs) were fabricated using a silicone foam matrix and characterized under oscillatory shear loading. SEM was used to quantify cellular morphology and effective particle volume fraction, while strain and frequency sweep tests were performed under magnetic flux densities from 0 to 0.8 T. The SEM results showed that increasing CIP content from 50 to 60 wt.% reduced the mean pore diameter from 274.34 to 203.07 µm and decreased the void fraction from 67.70% to 63.62%. Frequency sweep results showed that the storage modulus increased with both frequency and magnetic field, while the loss modulus exhibited a broad relaxation peak and increased in magnitude under applied field. A novel model integrating phenomenological and mechanistic approaches was subsequently developed by combining a two-branch generalized Maxwell model with a magnetic dipole interaction-based stiffening term, and a field-dependent magnetic loss branch. The model accurately reproduced the measured storage and loss moduli, with R2 values above 0.95 for G′ and 0.94–0.99 for G′′. The results demonstrate that MR foam behavior is governed by the coupled effects of foam morphology, effective particle volume fraction, excitation frequency, and magnetic field strength. The proposed model provides a compact and physically interpretable framework for predicting and designing tunable MR foams for multifunctional adaptive systems, including vibration isolation, impact mitigation, acoustic control, soft robotics, and lightweight aerospace structures.