The popularity of the studies is getting further on the flexomagnetic (FM) response of nano-electro-magneto machines. In spite of this, there are a few incompatibilities with the available FM model. This study indicates that the accessible FM model is inappropriate when considering the converse magnetization effect that demonstrates the necessity and importance of deriving a new FM relation. Additionally, the literature has neglected the converse FM coefficient in the Lifshitz invariant inside the free energy constitutive relation.

This fact inspires us to endeavor and conduct a new characteristic formulation for static analysis of axially compressed piezomagnetic nanobeams comprising the FM effect. This novel FM model is competent and suitable for various boundary conditions, encompassing analytical, semi-analytical, and numerical solving strategies. However, based on the previous FM equation established with respect to Euler-Bernoulli and Timoshenko beams, the governing equations are ill-posed due to the corresponding energy density. Despite that, this error will not remain in the finalized equations in the present model by conjecturing a gradient of the magnetic field and a different formulation. Moreover, the inverse FM parameter will appear in the magnetic field relation.

As the literature reported, non-uniform deformed piezomagnetic structures are capable of presenting more outstanding flexomagneticity. In actuality, a non-uniform elastic strain appears as a response to the magnetic field gradient (converse effect) that causes this study to deduce the nanobeam with higher-order shear deformations. Furthermore, the local governing equations will be transferred into the nonlocal phase according to the nonlocal differential, particularly nonlocal integral elasticity which itself is a strong nonlocality. Through this theory, and in regard to the converse FM impact, an analytical expression is applied for computing critical buckling loads within several ends conditions of the nanobeam. Our present results and achievements will hopefully be an effective contribution to theoretical studies on the mechanics of intelligent nanostructures.

纳米电磁电机的挠曲磁 (FM) 响应研究越来越受欢迎。尽管如此，与可用的 FM 型号仍有一些不兼容之处。这项研究表明，在考虑证明推导新 FM 关系的必要性和重要性的逆磁化效应时，可访问的 FM 模型是不合适的。此外，文献忽略了自由能本构关系中 Lifshitz 不变量中的逆 FM 系数。

这一事实激励我们努力并进行一种新的特征公式，用于对包含 FM 效应的轴向压缩压磁纳米束进行静态分析。这种新颖的 FM 模型能够胜任并适用于各种边界条件，包括解析、半解析和数值求解策略。然而，基于之前关于 Euler-Bernoulli 和 Timoshenko 梁建立的 FM 方程，由于相应的能量密度，控制方程是不适定的。尽管如此，通过推测磁场梯度和不同的公式，这个错误不会保留在当前模型的最终方程中。而且，在磁场关系中会出现反调频参数。

正如文献报道的那样，非均匀变形的压磁结构能够呈现出更出色的柔磁特性。实际上，不均匀的弹性应变表现为对磁场梯度（逆效应）的响应，导致本研究推断出具有高阶剪切变形的纳米束。此外，局部控制方程将根据非局部微分转移到非局部阶段，特别是非局部积分弹性本身就是一个强的非局部性。通过该理论，并针对相反的 FM 影响，应用解析表达式来计算纳米梁的几个末端条件内的临界屈曲载荷。