## 1. Introduction

To study the flexomagnetic (FM) effect and to better identify it, one can use the family close to it, that is, the piezomagnetic effect. In piezomagnetic, simply by compressing or stretching materials, an internal magnetic field is created in them. The piezomagnetic effect and its application can be seen in many materials and structures. However, in addition to these very useful applications, there is an important drawback that this effect can only exist in about 20 crystal structures with a specific symmetrical classification. However, there is no such limit to the FM effect, and materials with wider classes of symmetry can cause such a phenomenon. The flexomagnetic effect can be very strong and effective, so that it may one day be used in nanosensors or nanometer actuators. As a brief explanation of the FM effect, it can be noted that by bending an ionic crystal, the atomic layers are drawn inside it, and it is clear that the outermost layer will have the most tension. This difference in traction in different layers can cause ions to transfer to the crystal so much that they eventually create a magnetic field. In other words, bending some materials creates a magnetic field, a corresponding phenomenon called flexomagnetic effect. The effect of strain gradients shows that the importance of the FM effect in micro and nano systems is comparable to that of piezomagnetic and even beyond. Additionally, flexomagnetic, unlike piezomagnetic, can be found in a wider class of materials. This means that compared to piezomagnetic, which is invalid and inefficient in materials with central symmetry, there is an FM effect in all biological materials and systems. These traits have led to a growing interest in and research into the flexomagnetic effect in recent years [

1,

2]. Currently, the role of the flexomagnetic effect in the physics of dielectrics has been investigated in some studies and has shown promising practical applications [

3,

4,

5,

6,

7]. On the other hand, the difference between theoretical and experimental results shows a limited understanding in this field. This study examines current knowledge of FM in engineering.

The flexomagnetic effect exists in many solid dielectrics, soft membranes, and biological filaments. The flexomagnetic effect is introduced as the effect of size-dependent electromagnetic coupling due to the presence of strain gradients and magnetic fields, and promises many applications in nano-electronic devices (with strong strain gradients). Just as the piezomagnetic effect is expected to have important applications in nano-engines and particles [

8,

9,

10,

11,

12], so the FM effect can play this role as well. Different fields of science are used to study nanodielectrics by considering the FM effect. These significant parts can be examined from a chemistry and physics point of view, or they can be put under a magnifier in the engineering and industrial aspects. In the engineering aspects, the study of external factors on dielectrics and their mechanical and physical behavioral responses will naturally be the criterion for evaluation. The purpose of this study is to evaluate this aspect in static large deflection analysis of a nano actuator beam. A close look at the history of the study of the mechanical behavior of dielectrics by including the FM effect does not show many studies [

13,

14,

15]. These studies have generally looked at small deformations (linear strains), which, while important, cannot be the criterion for designing dielectric nanobeams. Definitely, the deformations should be considered as large as possible to obtain a reasonable and reliable safety factor for optimizing these significant nano-electro-magneto-mechanical systems’ components.

The present work accounts for the large deflections by adding the nonlinear terms of Lagrangian strain using the von Kármán approach. The constitutive equations are expanded in line with the classical beam theory. It is worth mentioning that the small scale is fulfilled conforming to the second stress and strain gradients. These extra terms should result in two conflict responses, that is softening and hardening in the nanoscale structure based on the literature. We perform the solution of acquired equations, which govern the nonlinear bending of the nanobeam, on the basis of two step solution techniques. The first one is the Galerkin weighted residual method (GWRM) which converts the equations into nonlinear algebraic ones, then the Newton–Raphson technique (NRT), which solves the nonlinear system of algebraic equations and gives the numerical values of displacements into x and z directions. At last, pictorial results are evaluated to show the disagreements and dissimilarities betwixt linear deflection and nonlinear one for the piezo-flexomagnetic nanosize beam.

## 2. Mathematical Model

Let us consider a piezomagnetic-flexomagnetic nanobeam (PF-NB) with squared cross section of length and thickness

L and

h; see

Figure 1. A uniform vertical static loading acts above the beam. A magnetic potential is joint to the beam to simulate and act as a magnetic field. Moreover, the

z-axis is related to the transverse direction, whereas the neutral plane of the beam is coincident with the

x-axis.

Follow up, the kinematic displacement for each node of the beam is utilized with the aid of the Euler–Bernoulli hypothesis [

16,

17]. Furthermore, the model is restricted with in-plane deformations. The rectangular displacements correspond with

u_{1} and

u_{3}, respectively, for axial and transverse directions. However, such displacements for neutral plane are, respectively, regarded with

u and

w. Thus, one can give accordingly

The Von Kármán assumption tells us that the nonlinear terms related to the

u can be excluded from the Lagrangian strain formula because these terms are sufficiently small compared to the other terms [

18,

19,

20,

21,

22,

23,

24]. The general Lagrangian strain can be mentioned as

In regard to this approach, the nonzero nonlinear strain-displacement components can be derived as follows

where Equations (4) and (5) calculate, respectively, the longitudinal strain and its gradient.

The stress-strain magneto-mechanical coupling relations in the one-dimensional framework can be given owing to [

13,

14].

where

${\sigma}_{xx}$ is the static stress field component,

${H}_{z}$ is the magnetic field component,

${B}_{z}$ is the magnetic flux (induction) component,

${C}_{11}$ is the elastic modulus,

${f}_{31}$ is the component of the fourth-order flexomagnetic coefficients tensor,

${a}_{33}$ is the component of the second-order magnetic permeability tensor,

${q}_{31}$ is the component of the third-order piezomagnetic tensor,

${g}_{31}$ is the component of the sixth-order gradient elasticity tensor, and

${\xi}_{xxz}$ is the component of higher-order moment stress tensor.

The variational formulation accurately develops the characteristics relation of PF-NB, thusly

where

$\delta $ is the symbol of variation,

$U$ is the strain energies, and

$W$ is created works by outer objects. In such a way, the entire inner energy of the specimen is in the first variation which is equal to zero as well. The strain energy respecting magneto-mechanical composition can be variated just like this (the first variation)

Equation (10) can be transformed with integration by parts on the basis of the one-dimensional displacement field previously assumed as follows

where

where

$\mathsf{\Psi}$ is the variable of magnetic potential. The resultants of the stress field can be introduced along the following lines

In addition, the magnetic potential was introduced through the relation

External forces (axial force as a result of the longitudinal magnetic field and the lateral loading) create work thermodynamically in the particles so that the mathematical relation in the first variation becomes [

25].

in which

${N}_{x}^{0}$ is the in-plane longitudinal axial force, and

$p$ is the lateral load per unit length. Taking into account the closed circuit in conjunction with the inverse piezo case, the electrical boundary conditions can be attributed as below

in which

$\psi $ is the external magnetic potential on the upper surface. Making in hand Equations (8), (13), (15), (21) and (22) practicably expresses the magnetic field component and thereupon the magnetic potential function in line with thickness as follows [

13,

14]

On the basis of Equations (23) and (24), Equations (6)–(8) can be developed as

Subsequently, Equations (16)–(18) can be rewritten in detail as

in which

${N}_{x}^{},\text{}{M}_{x}^{},\text{}{T}_{xxz}$ show the axial, moment, and hyper stress resultants, and

${I}_{z}^{}={\displaystyle {\int}_{A}{z}^{2}dA}$ is the area moment of inertia.

The resultant magnetic axial stress, which is achieved due to the longitudinal magnetic field, based on Equation (28) can be determined as

This force is supposed to act at both ends of the beam, thus

Eventually, imposing Equation (9), one can write the governing equations in a combination of mechanical and magnetic conditions as

Due to being the nanobeam a size-dependent particle, the scale-dependent property should be substituted in Equations (33) and (34). In [

26], the second strain gradient of Mindlin merged successfully with the nonlocal theory of Eringen. This model (NSGT) was incorporated in a lot of research performed on the nanoparticles in recent years—see e.g., [

27,

28,

29,

30,

31,

32,

33,

34,

35,

36,

37,

38] and many others—and can be a proper item at the nanoscale.

The model proposed by [

26] can be compatible in our case as

or as

in which

$\mu \left(n{m}^{2}\right)$ is the nonlocal parameter, and

$l\left(nm\right)$ is the strain gradient parameter. Thus,

$l>0$ establishes a nonzero strain gradient into the model, and

$\mu ={\left({e}_{0}a\right)}^{2}$ is the parameter defining nonlocality. It is germane to note that both scale parameters are dependent on the physics of the model and cannot be material constants [

39,

40]. This means the parameters are not constant values, something like an elasticity modulus for each material.

To implement the influence of size effects into the equations, Equation (35) is plugged to Equations (28)–(30) as

Equations (33) and (34) by means of Equations (36)–(38) can be derived in the framework of displacements, respectively, as series of models.

1.1. Piezo-flexomagnetic nanobeam (PF-NB)—Nonlinear case:

1.2. Piezo-flexomagnetic nanobeam (PF-NB)—Linear case:

2.1. Piezomagnetic nanobeam (P-NB)—Nonlinear case:

2.2. Piezomagnetic nanobeam (P-NB)—Linear case:

3.1. Nanobeam (NB)—Nonlinear case:

3.2. Nanobeam (NB)—Linear case:

4.1. Classic beam—Nonlinear case:

4.2. Classic beam—Linear case:

In what follows, we consider these cases in more details.

## 3. Solution Approach

The solution process here has two steps. The first step comes with the Galerkin weighted residual method (GWRM) on the basis of the admissible shape functions which satisfy boundary conditions. The second step is imposing the Newton–Raphson technique (NRT) in order to solve the system of nonlinear algebraic equations originated from GWRM. The following displacements were employed [

41].

where

${U}_{m}$ and

${W}_{m}$ are unknown variables that determine displacements through two axes and should be computed, whereas

${X}_{m}\left(x\right)$ are shape functions,

m is the axial half-wave number, and becomes

$m=1,2,\dots \infty $. The allowable shape functions given below satisfy end conditions as [

41].

in which S, C, and F mark one by one the simply-supported, clamped, and free end conditions. Here, e.g., C-F means a side of the beam is inserted in a clamping fixture and the opposite side is free and hanging.

Based on the Fourier sine series, the transverse load can uniformly behave on the nanobeam as the following form [

42,

43].

in which

${p}_{0}$ is density of the lateral load. Inserting Equations (51), (52), and (56) into Equations (39)–(50), and integrating over the axial domain based on the GWRM approach, one can obtain

in which

$\eta $ and

$\xi $ are the first and second equations, respectively, and

${Y}_{m}$ and

${Z}_{m}$ show the residuals. Then, with ordering and arranging the aforesaid equations, one can receive the nonlinear algebraic system of two equations and two unknown variables (when considering

m = 1). To solve such a system, there are several methods. As long as the NRT converged the results very quickly and accurately, this technique was employed here. A primary guess (

${U}_{0}$ and

${W}_{0}$) was required for results in this approach. We can express the first iteration as [

44].

where

J denotes the Jacobian matrix 2 × 2 and

A is a vector 2 × 1.

where

e is the governing equations with placing the first guesses. As a matter of fact, Equations (59) and (60) are iterative equations that are

where

n is the number of iterations to receive the convergence. A few iterations are enough to obtain the desired accuracy. It is worth mentioning that the convergence and the expected accuracy were completely dependent on the value of the primary guesses. Consequently, the solution led to numerical values of displacements along axial and transverse axes. To plot the results for large deflections, we needed to obtain the vertical displacement only, and the other will not be drawn.