Nonlinear free vibration of magneto–electro–elastic sandwich plates with auxetic cores under coupled thermo–electro–magnetic fields

This study presents a comprehensive nonlinear vibration analysis of magneto–electro–elastic (MEE) sandwich plates incorporating auxetic core architectures under coupled thermo–electro–magnetic fields. A semi-analytical formulation based on higher-order shear deformation theory (HSDT) and nonlocal strain-gradient theory is developed to capture geometric nonlinearity, size dependency, and multi-field coupling effects. Finite element analysis is employed only at the unit-cell level to homogenize the effective elastic properties of the anti-tetrachiral auxetic core. The results reveal that increasing the auxetic core thickness ratio leads to a reduction of the fundamental normalized frequency by approximately 15–35%, depending on the geometric configuration, due to enhanced internal rotational mechanisms and reduced effective bending stiffness. Thermal loading is found to have a dominant influence on the dynamic response, causing a continuous decrease in frequency and leading to a critical minimum at ΔT ≈ 2000 K, where the frequency drops by nearly 40–60% compared to the initial state. This temperature level may be interpreted as a critical-like thermally dominated transition region, where thermally induced compressive stresses strongly reduce the effective tangent stiffness; beyond this region, geometric nonlinearity contributes to a partial stiffness recovery. Furthermore, the material length-scale parameter significantly enhances stiffness at small scales, increasing the frequency by up to 20% in the low-temperature regime, while its influence diminishes near the frequency-minimum temperature. The applied magnetic potential exhibits a stabilizing effect, increasing the frequency by approximately 10–25%, whereas the electric potential induces a softening behavior and shifts the frequency-minimum temperature to lower values, accelerating instability. In addition, the incorporation of an elastic foundation is shown to significantly improve structural stability, with increasing foundation stiffness suppressing thermal softening and delaying the onset of instability. Overall, the study demonstrates that the nonlinear vibration behavior of auxetic-core MEE sandwich plates is governed by a strong competition between thermal softening and multi-field-induced stiffness modulation. The findings provide new physical insight and practical guidelines for the design of advanced smart structures operating under severe multi-field environments.