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

Abstract

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.

On nonlinear dilatational strain gradient elasticity

“…Here, h is the mean curvature of a, h = −1/2∇ s • n, η is the outward normal vector to ∂a such that η • n = 0. With (77), one obtains the formula…”

“…[14,56,57]; (v) damage and fracture phenomena [100,101,103,111,124]; (vi) growth of bone tissues and its effects on bone mechanical properties [60,61,76,118]; (vii) coupled electromechanical phenomena as flexoelectricity and flexomagneticity, see, e.g. [37,46,77,78,80,132,136] and the reference therein;…”

Critical Temperatures for Vibrations and Buckling of Magneto-Electro-Elastic Nonlocal Strain Gradient Plates

“…In [45] are studied the pyroelectric and pyromagnetic effects on multiphase MEE cylindrical shells subjected to a uniform axisymmetric temperature using semi-analytical finite element procedures. Eremeyev et al in [46,47] investigate the effect of flexoelectricity and flexomagnetic on nanobeams. Using FEM formulation [48,49] analyzed free vibration of orthotropic cross-ply nanoplates and nanowires.…”

Effect of Axial Porosities on Flexomagnetic Response of In-Plane Compressed Piezomagnetic Nanobeams

“…In discussing a mechanical response of nanomaterials with magneto-mechanical coupling, the importance of piezomagneticity has been profoundly evaluated by scholars in the contemporary decade [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. However, fewer studies are available regarding flexomagneticity [40][41][42][43][44][45]. In the available literature, Sidhardh and Ray [40] worked on the bending response of a thin cantilever nanobeam with flexomagnetic property.…”

“…The small scale response of the nanoscale beam was searched by imposing a nonlocal strain gradient elasticity approach into the constitutive equations. Besides these, Malikan and Eremeyev [44] performed research on non-linear static bending of smart nanoscale beams while the material included a remarkable flexomagnetic response. The computational model was solved by coupling between an analytical and numerical solution method.…”