On instabilities and post-buckling of piezomagnetic and flexomagnetic nanostructures

Nonlinear eigenfrequency characteristics of multi-directional functionally graded porous panels

On the geometrically nonlinear vibration of a piezo‐flexomagnetic nanotube

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.

https://www.mdpi.com/2079-4991/10/9/1762/pdf

On Nonlinear Bending Study of a Piezo-Flexomagnetic Nanobeam Based on an Analytical-Numerical Solution.

Based on the Timoshenko beam theory, von Karman geometric nonlinearity assumption and the modified couple stress theory, this paper presents linear and nonlinear free vibration analysis of rotating two-dimensional functionally graded micro-beam with even and uneven porous distributions. The material properties vary along the axial and thickness directions. The NURBS function is employed as the basic one to overcome the inherent difficulty of constructing higher order continuous displacement function for modified couple stress theory based finite element formulation. The validity of the present simulation is validated by comparing numerical results with those available literatures for rotating functionally graded micro-beams. Finally, the effects of several parameters such as porosity, power law index, material length scale parameter, rotational speed and hub radius on dimensionless frequencies and nonlinear frequency ratios are discussed.

Nonlinear free vibration analysis of a rotating two-dimensional functionally graded porous micro-beam using isogeometric analysis

The present paper presents a Bernoulli–Euler flexomagnetic (FM) nanobeam model, which considers the effects of flexomagneticity, piezomagneticity, and the surface elasticity. Differential control equations and corresponding magnetic boundary conditions are derived to investigate the influences of direct and converse FM couplings over the magnetic-elastic response. Size-dependent theoretical solutions for the static bending deformation of the cantilever, simply supported, and clamped nanobeams subjected to concentrated or uniformly distributed load are derived. Numerical simulations demonstrate that the flexomagneticity effect plays the role of the scale-dependent enhancement of the bending rigidity, which is independent of boundary conditions. But for the residual surface stresses, softening or stiffening the beam depends on boundary conditions.

Dejin Chen

Papers

Nonlinear free vibration analysis of a rotating two-dimensional functionally graded porous micro-beam using isogeometric analysis

Size-dependent static bending of flexomagnetic nanobeams