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

Mohammad Malikan,Krzysztof Żur,Victor Eremeyev

doi:10.3390/sym12121935

Abstract

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.

Highlights

Flexomagneticity arises through elastic strain gradient or magnetic field gradient during electric magnetization in the magneto-elastic coupling in smart structures and actuators [1,2,3]
The present research attempts to demonstrate the flexomagnetic property for the stability problem of a nano-sized beam, while it includes a material imperfection with intentional nonlocality and size-dependent characteristics according to the nonlocal strain gradient constitutive equation
In which ηxxz and εxx are the gradient of the axial elastic strain and the strain itself, C11 = C1111 is the elastic modulus, σxx is the axial stress, f31 = f3311 denotes the component of the fourth-order flexomagnetic coefficients tensor, a33 represents the component of the second-order magnetic permeability tensor, q31 = q311 depicts the component of the third-order piezomagnetic tensor, ξxxz is the component of the higher-order hyper stress tensor and is an induction of coverse flexomagnetic effect, Bz and Hz exhibit the magnetic flux and the component of magnetic field, respectively, and g31 = g311311 illustrates the influence of the sixth-order gradient elasticity tensor

Summary

Introduction

Flexomagneticity arises through elastic strain gradient or magnetic field gradient during electric magnetization in the magneto-elastic coupling in smart structures and actuators [1,2,3]. Sidhardh and Ray [40] worked on the bending response of a thin cantilever nanobeam with flexomagnetic property They discussed both direct and reverse impacts of magneto-elastic coupling with the presence of the surface elasticity. To present the size-dependent mechanical behavior of the structure, they utilized the surface elasticity Both direct and converse flexomagnetic influences were investigated when the nano-sized beam was kept in ends with fixed, pivot, and free edge conditions. Malikan and Eremeyev [44] performed research on non-linear static bending of smart nanoscale beams while the material included a remarkable flexomagnetic response. The present research attempts to demonstrate the flexomagnetic property for the stability problem of a nano-sized beam, while it includes a material imperfection with intentional nonlocality and size-dependent characteristics according to the nonlocal strain gradient constitutive equation. The nanobeam is considered for variations of key parameters based on the three cases, i.e., a simple nanobeam, piezomagnetic nanobeam, and piezo-flexomagnetic nanobeam

Constitutive Relations for Piezo-Flexomagnetic Solids

The Piezo-Flexomagnetic Beam Model

Solution of the Problem

Validation of Results

Stability Analysis

Conclusions