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In this paper, the equation governing the buckling of a magnetic composite nanoplate under the influence of an in-plane one-dimensional magnetic field, assuming the concept of flexomagnetic and considering the resulting flexural force and moment, is investigated for the first time by different analytical boundary conditions. To determine the equation governing the stability of the nanoplate, the nonlocal strain gradient theory has been used by taking into account the classical plate theory. The axial magnetic force, which is originated from the magnetic field, is investigated. After extracting the governing differential equation, the critical buckling load is obtained for different support conditions. The effect of nonlocal parameter, sheet aspect ratio and the effect of one-dimensional magnetic field on critical load are discussed. It was earned that if the nanoplate is rectangular so that the value of aspect ratio is less than one, the flexomagnetic response will be more noticeable. [ABSTRACT FROM AUTHOR]

We investigated the stability of an axially loaded Euler–Bernoulli porous nanobeam considering the flexomagnetic material properties. The flexomagneticity relates to the magnetization with strain gradients. Here we assume both piezomagnetic and flexomagnetic phenomena are coupled simultaneously with elastic relations in an inverse magnetization. Similar to flexoelectricity, the flexomagneticity is a size-dependent property. Therefore, its effect is more pronounced at small scales. We merge the stability equation with a nonlocal model of the strain gradient elasticity. The Navier sinusoidal transverse deflection is employed to attain the critical buckling load. Furthermore, different types of axial symmetric and asymmetric porosity distributions are studied. It was revealed that regardless of the high magnetic field, one can realize the flexomagnetic effect at a small scale. We demonstrate as well that for the larger thicknesses a difference between responses of piezomagnetic and piezo-flexomagnetic nanobeams would not be significant. [ABSTRACT FROM AUTHOR]

Among various magneto-elastic phenomena, flexomagnetic (FM) coupling can be defined as a dependence between strain gradient and magnetic polarization and, contrariwise, elastic strain and magnetic field gradient. This feature is a higher-order one than piezomagnetic, which is the magnetic response to strain. At the nanoscale, where large strain gradients are expected, the FM effect is significant and could be even dominant. In this article, we develop a model of a simultaneously coupled piezomagnetic–flexomagnetic nanosized Euler–Bernoulli beam and solve the corresponding problems. In order to evaluate the FM on the nanoscale, the well-known nonlocal model of strain gradient (NSGT) is implemented, by which the nanosize beam can be transferred into a continuum framework. To access the equations of nonlinear bending, we use the variational formulation. Converting the nonlinear system of differential equations into algebraic ones makes the solution simpler. This is performed by the Galerkin weighted residual method (GWRM) for three conditions of ends, that is to say clamp, free, and pinned (simply supported). Then, the system of nonlinear algebraic equations is solved on the basis of the Newton–Raphson iteration technique (NRT) which brings about numerical values of nonlinear deflections. We discovered that the FM effect causes the reduction in deflections in the piezo-flexomagnetic nanobeam. [ABSTRACT FROM AUTHOR]