This research work performs the first time exploring and addressing the flexomagnetic property in a shear deformable piezomagnetic structure. The strain gradient reveals flexomagneticity in a magnetization phenomenon of structures regardless of their atomic lattice is symmetrical or asymmetrical. It is assumed that a synchronous converse magnetization couples both piezomagnetic and flexomagnetic features into the material structure. The mathematical modeling begins with the Timoshenko beam model to find the governing equations and non-classical boundary conditions based on shear deformations. Flexomagneticity evolves at a small scale and dominant at micro/nanosize structures. Meanwhile, the well-known Eringen’s-type model of nonlocal strain gradient elasticity is integrated with the mathematical process to fulfill the scaling behavior. From the viewpoint of the solution, the displacement of the physical model after deformation is carried out as the analytical solution of the Galerkin weighted residual method (GWRM), helping us obtain the numerical outcomes on the basis of the simple end conditions. The best of our achievements display that considering shear deformation is essential for nanobeams with larger values of strain gradient parameter and small amounts of the nonlocal coefficient. Furthermore, we showed that the flexomagnetic (FM) effect brings about more noticeable shear deformations’ influence.

这项研究工作首次探索和解决了剪切可变形压电结构中的柔磁特性。应变梯度揭示了结构磁化现象中的挠曲磁性，而不管它们的原子晶格是对称的还是不对称的。假设同步逆磁化将压电和挠磁特征耦合到材料结构中。数学建模从 Timoshenko 梁模型开始，以找到基于剪切变形的控制方程和非经典边界条件。柔性磁性在小尺度上发展并在微/纳米尺寸结构中占主导地位。同时，著名的非局部应变梯度弹性的 Eringen 型模型与数学过程相结合，以实现缩放行为。从解的角度来看，物理模型变形后的位移作为伽辽金加权残差法（GWRM）的解析解进行，有助于我们在简单的终端条件的基础上得到数值结果。我们的最佳成果表明，考虑剪切变形对于具有较大应变梯度参数值和少量非局部系数的纳米梁至关重要。此外，我们表明挠磁 (FM) 效应带来了更明显的剪切变形影响。我们的最佳成果表明，考虑剪切变形对于具有较大应变梯度参数值和少量非局部系数的纳米梁至关重要。此外，我们表明挠磁 (FM) 效应带来了更明显的剪切变形影响。我们的最佳成果表明，考虑剪切变形对于具有较大应变梯度参数值和少量非局部系数的纳米梁至关重要。此外，我们表明挠磁 (FM) 效应带来了更明显的剪切变形影响。