An Analytical Approach for Simulating the Bending of Nanobeams in Thermal Environments Considering the Flexomagnetic Effect

1  Introduction

Currently, smart materials are widely used in several significant sectors such as aerospace, medical, military industry, and others. Common examples of smart materials include materials that exhibit varying mechanical characteristics, piezoelectric materials, magnetic materials, materials with flexoelectric effects, materials with flexomagnetic effects, and so on. Therefore, the investigation into the mechanical behavior of structures composed of this intelligent material has taken a substantial leap ahead. Zhang et al. [1] performed research on the bending characteristics of piezoelectric material plates that have a bidirectional functionally graded (FG) structure inside the plane. Jing et al. [2] conducted an oscillation evaluation of piezoelectric cantilever beams with bimodular FG characteristics. Khiem et al. [3,4] completed a study on the vibration response of a cracked beam made of functionally graded material (FGM) with a piezoelectric section, subjected to a moving load. Li and collaborators [5] conducted a study on the electro-mechanical vibration and stress distribution of a piezoelectric nanobeam with a symmetrical FGM core when subjected to a low-velocity reaction. Binh et al. [6] demonstrated the use of analytical modeling to analyze the mechanical properties of piezoelectric nanobeams, focusing on the inclusion of flexoelectric influences. Son and coworkers [7] proposed a theoretical framework called the novel shear deformation theory. This theory aims to accurately model the behavior of nanoplates under dynamic conditions. Doan et al. [8] investigated the impact of the flexoelectric effect on the free oscillation characteristics of cracked nanoplates using the phase-field theory. Askari and colleagues [9] presented a study of the vibrations in beams that are both transversely and sheared linked and made of piezoelectric functionally graded porous materials. The analysis was conducted using higher-order theories. Duc and Phuc [10] presented a thorough investigation of the mechanical behavior of piezoelectric nanoplates with varying thicknesses, including the influence of the flexoelectric effect.

Small-scale electronics, such as nanosensors, transistors, nanogenerators, and actuators, frequently consist of beam or nanoplate structures that experience various forms of stress. Research has successfully shown their mechanical reactions. Jankowski et al. [11] discussed the piezoelectric influence on the bifurcation buckling of a symmetric FGM porous nanobeam. This was done using a higher-order nonlocal elasticity and strain gradient theory, together with Reddy’s third-order shear deformation beam theory. Sator et al. [12] delivered a presentation on the analysis of FG piezoelectric micro/nano plates using the moving finite element method (FEM). Kammoun and coworkers [13] researched the study of thermo-electro-mechanical dynamics and free vibrations of a nanobeam made of functionally graded piezoelectric material, taking into account the effects of initial stress. Van Thom and colleagues [14] conducted a study on the mechanical reactions of nanoplates when placed on viscoelastic foundations in settings with multiple physical properties. Hoan et al. [15] demonstrated nonlinear vibration nanobeam analysis sustaining thermal environment with flexomagnetic effect consideration. Selvamani and collaborators [16] examined the propagation of nonlocal state-space strain gradient waves in an FG magnetothermoelectric nanobeam. Ebrahimi et al. [17] scrutinized the Influence of temporal nonlocality on the wave propagation characteristics of viscoelastic FGM nanoshells. Li and his team [18] explored the dynamic behaviors of imperfect functionally graded piezoelectric sandwich microplates with thermal-electric properties. The study was based on the modified couple stress theory. Wang and coworkers [19] delivered a study on the examination of wave propagation in porous FG piezoelectric nanoshells that are supported by a viscoelastic foundation. Fang et al. [20] and Tien et al. [21] presented an application of the nonlocal hypothesis to analyze the free and forced oscillation of multilayer nanoplates. Phung [22] delivered a presentation on the static bending analysis of nanoplates on a discontinuous elastic foundation, focusing specifically on the inclusion of the flexoelectric effect. Sun et al. [23] conducted a study on the characteristics of nonlinear dynamical instability in FG piezoelectric microshells. The study included the effects of nonlocality and strain gradient size dependencies. The most recent related studies can be found in these documents [24–30]. Lan et al. [24] combined nonlocal elasticity with the Galerkin method, Tho et al. [25] adopted nonlocal elasticity within an analytical framework, and Dehsaraji et al. [26] integrated higher-order shear deformation theory with nonlocal theory. Yademellat et al. [27] employed both classical beam theory and the nonlocal strain gradient theory, while Cong et al. [28] utilized a phase-field approach to examine the influence of size effects on crack propagation in structural components. In addition, for structures at the nanoscale, the surface effect is one of the key factors significantly influencing their mechanical responses [31–33]. However, when analyzing nanoscale structures, various theoretical models have been proposed, among which Eringen’s model [34] is one of the most widely used. However, this model can lead to inaccuracies under certain special boundary conditions. Therefore, researchers have developed more advanced and effective models for solving nanoscale structural mechanics problems [35–39]. Romano et al. [35] examined the linear problem, Ussorio et al. [37] investigated the nonlinear case, while Vaccaro et al. [38,39] developed higher-order theories to improve the accuracy of the analysis. In this work, the proposed Eringen model [34] is adopted, as it is simple, computationally convenient, and still widely accepted and employed by many scholars [21,24,25,31,32].

Beam constructions can endure several sorts of loads, including mechanical loads and thermal stresses. Due to this factor, beams can experience significant deformations. In addition, the presence of the flexomagnetic effect further complicates the beam’s reaction, resulting in an entirely distinct behavior compared to situations involving just minor deformations. There are various methods available for beam analysis. For example, Mahboubi Nasrekani and Eipakchi [40], Chandel and Talha [41] employed the perturbation approach, Wu [42] applied the Meshless method, Useche and Alvarez [43] adopted the Boundary Element Method, while Pereira et al. [44] and Liu et al. [45] utilized approximate solution techniques. Unlike other approximate methods [40,46,47], the objective of this research is to develop an innovative approach based on analytical methodologies. The presented findings will undoubtedly provide designers with a foundation for developing efficient structures that fulfill the specified criteria.

This paper is organized into the following primary sections: The equilibrium equation and solution for the issue of nonlinear bending of nanobeams are described in Section 2. Section 3 presents some instances to show the dependability of the solution in this study. Section 4 presents the outcomes of numerical computations and the subsequent examination of these results. Section 5 provides a concise overview of the primary findings that have been examined in this study.

2  Nano Beams Subjected to Bending, Considering Geometric Nonlinearity and Flexoemagnetic Effects

Fig. 1 illustrates the nanobeam model constructed from materials exhibiting flexomagnetic properties. Align the x-axis parallel to the beam axis, and position the z-axis perpendicular to the beam axis. The beam has a length denoted as L, a height represented by h, and is situated inside an environment that experiences mechanical stress Fz and temperature variations.

Figure 1: Model of a nanobeam with flexomagnetic effect in a temperature environment

There are several theories available for calculating beams. However, when it comes to nonlinear issues, most projects have relied on classical beam theory. This is due to the simplicity and convenience of its structure, which facilitates calculations. However, the drawback of the classical theory lies in its limited applicability to slender beams. When dealing with beams that have a significant thickness, using this theory for calculations will result in substantial inaccuracies since it fails to account for the impact of shear deformation. This study employs the novel third-order shear deformation theory to provide a solution to the bending issue of nanobeams affected by the flexomagnetic effect. The two displacement components are expressed in the following manner [48]:

{χx=χ0x−z∂χb∂x−gz∂χs∂xχz=χb+χs(1)

where χx and χ0x are the axial displacements along the x-axis at the position with coordinate z and at the position on the midline of the beam, respectively. The displacement χz is measured perpendicular to the beam axis, which is aligned with the z-axis. The distinction between this novel form of higher-order theory and the traditional theory lies in its incorporation of the χs component, which accounts for shear displacement and shear strain, and

gz=(127z4200h3−77z2125h+39h500)sinh⁡(πzh)(2)

When employing the function gz as expressed in Eq. (2), although the thickness integration becomes more complex compared to other functions, the use of this gz function ensures that the shear stresses vanish at the beam surfaces without the need for any shear correction factors, while accurately capturing the mechanical response of the structure.

After the displacement field derivative, the strains consider the large deformation:

{εx=ε0+εTh+zεz+gzεgΓxz=fzΓ0xz(3)

in which the axial strain takes into account the influence of temperature:

ε0=∂χ0x∂x+12(∂χz∂x)2;εz=−∂2χb∂x2;εg=−∂2χs∂x2εTh=αTherΔT(4)

where αTher is the coefficient of thermal expansion, and ΔT is the deviation of the ambient temperature from the stress-free temperature of the material, and shear strain is as follows:

Γxz=∂χs∂x;fz=1−∂gz∂z(5)

This study examines the effects of significant deformation and thermal deformation caused by temperature changes. This augments the intricacy and unwieldiness of the computation formulae provided below. However, addressing the issue of nonlinear bending in nanobeams using this theory will be more significant since the theoretical calculation model closely aligns with real-world conditions.

Continuing to differentiate the strain, one derives the strain gradient:

ηxxz=−∂2χb∂x2−∂gz∂z∂2χs∂x2=−∂2vb∂x2+∂gz∂z(−∂2χs∂x2)=ηb+∂gz∂zηs(6)

where ηb=−∂2vb∂x2;ηs=−∂2χs∂x2.

This study considers only the size-dependent effect based on nonlocal theory, without accounting for the influence of the surface effect. The flexomagnetic effect and the size effect make the relationship between stress and strain as [20,28]:

(1−Γ2∂2∂x2)σx=Eεx−q31Vz(7)

(1−Γ2∂2∂x2)τxz=Gεxz(8)

Ψxxz=g14ηxxz−k14Vz(9)

Pz=q31εxx+κ33Vz+k14ηxxz(10)

in which σx and τxz are the normal stress and the tangent stress, Γ is a parameter representing the size effect, Vz is the magnetic field, q31 is the piezomagnetic coefficient, k14 is the flexomagnetic coefficient, Ψxxz is the higher-order stress, κij is the magnetoelectric coefficient, và Pz is the magnetic flux, G=E/2(1+ν), and g14 = λ3 + 2 λ4, where:

λ3=83λ2+25λ5;λ4=μ3(l12+6l22);λ2=μ30(27l02−4l12−15l22);λ5=μ3(l12−3l22)(11)

with μ is Lamé parameter, l0, l1, l2 are the material parameters.

The following can be obtained by applying Gauss’s law:

∂Pz∂z=0(12)

Using the condition Pz to vanish at the top and bottom surfaces of the cross-section, the magnetic field expression is now:

Vz=q¯31d33{∂2χb∂x2z+gz∂2χs∂x2}+k14d33∂gz∂z∂2χs∂x2+k14d33∂2χb∂x2(13)

Vertical force (Nx), bending moment (Mx), higher order moment (Sx), and shear force (Qxz) are calculated as follows:

Nx−Γ2∂2∂x2Nx=∫−h/2−h/2(c11εx−q31Vz)dz=A0(ε0+εTh)−q31k14hκ33εz−q31k14κ33ANgεg(14)

Mx−Γ2∂2∂x2Mx=∫−h/2h/2(c11εx−q31Vz)zdz=Azzεz+Azgεg−q312κ33h312εz−q312κ33AMzgεg(15)

Sx−Γ2∂2∂x2Sx=∫−h/2h/2(c11εx−q31Vz)gzdz=Agzεz+Aggεg−q312κ33AMzgεz−q312κ33ASggεg(16)

Qxz−Γ2∂2∂x2Qxz=∫−h/2h/2(GΓxz)dz=AsΓxz(17)

with the coefficients provided in the Appendix A.

Presently, the work principle is applicable to nanobeams:

δΘNB=δΘML(18)

where is the virtual work of the internal force of the nanobeam:

δΘNB=∫S∫−h/2h/2(σxδ(εx)+Ψxxzδηxxz+(τxz)δ(Γxz))dzdS=∫S(Nxδε0+Mxδεz+Sxδεf+Txxzδηb+Yzzxηs+QxzδΓxz)dS=∫S(Nx(δ∂χ0x∂x+∂χz∂xδ(∂χz∂x))−Mxδ(∂2χb∂x2)−Sxδ(∂2χs∂x2)−Txxzδ(∂2χb∂x2)−Yzzxδ(∂2χs∂x2)+Qxztzδ(∂χs∂x))dS=∫S(∂Nx∂xδχ0x+∂∂x(Nx∂χ∂x)δv−∂2Mx∂x2δχb−∂2Sx∂x2δχs−∂2Txxz∂x2δχb−∂2Yzzx∂x2δχs+∂Qxz∂xδχs)dS(19)

in which

Txxz=∫−h/2h/2Ψxxzdz=∫−h/2h/2{g14ηxxz−k14Vz}dz=∫−h/2h/2{g14ηb+g14∂gz∂zηs−k14(q¯31d33{∂2χb∂x2z+gz∂2χs∂x2}+k14d33∂gz∂z∂2χs∂x2+k14d33∂2χb∂x2)}dz=−[∫−h/2h/2(g14+k142d33)dz]∂2χb∂x2−[∫−h/2h/2(g14+k142d33)∂gz∂zdz]∂2χs∂x2=−ATb∂2χb∂x2−ATs∂2χs∂x2(20)

Yzzx=∫−h/2h/2Ψxxz∂gz∂zdz=∫−h/2h/2{g14ηxxz−k14Vz}∂gz∂zdz=∫−h/2h/2{g14∂gz∂zηb+g14(∂gz∂z)2ηs−k14(q¯31d33{∂2χb∂x2z∂gz∂z+gz∂gz∂z∂2χs∂x2}+k14d33(∂gz∂z)2∂2χs∂x2+k14d33∂gz∂z∂2χb∂x2)}dz=−[∫−h/2h/2(g14+k142d33)∂gz∂zdz]∂2χb∂x2−[∫−h/2h/2(g14+k142d33)(∂gz∂z)2dz]∂2χs∂x2=−AYb∂2χb∂x2−AYs∂2χs∂x2(21)

with the coefficients ATb, ATs, AYb, AYs provided in the Appendix A.

Virtual work of the mechanical force acting on the beam:

δΘML=∫SPvδχzdS(22)

Therefore, Eq. (18) is implemented into three equations as follows:

δχ0x:∂Nx∂x=0(23)

δχb:Nx∂2χ∂x2−∂2Mx∂x2−∂2Txxz∂x2−Pv=0(24)

δχs:Nx∂2χ∂x2−∂2Sx∂x2−∂2Yxxz∂x2+∂Qxz∂x−Pv=0(25)

The vertical force (14) now has the following form:

Nx=∫−h/2−h/2(c11εx−q31Vz)dz=A0(∂χ0x∂x+12(∂χz∂x)2+εTh)+q31k14hκ33∂2χb∂x2+q31k14κ33ANg∂2χs∂x2(26)

Integrating the above expression once, one derives:

Nxx+Nconst=A0(χ0x+12∫0x(∂χ(Ψ)∂Ψ)2dΨ+εThx)+q31k14hκ33∫0x∂2χb(Ψ)∂Ψ2dΨ+q31k14κ33ANg∫0x∂2χs(Ψ)∂Ψ2dΨ(27)

Suppose the beam is simply supported at both ends: χ0x (x = 0) = χ0x (x = L) = 0, one gets Nconst = 0. The vertical force Nx is now calculated as follows:

Nx=A02L∫0L(∂χz∂x)2dx+q31k14hκ33L∫0L∂2χb∂x2dx+q31k14κ33LANg∫0L∂2χs∂x2dx+A0εTh(28)

Hence, significant deformation and temperature variations have a direct impact on the stress and longitudinal force of the beam, resulting in a much more complex mathematical expression. The second derivative of the moment is obtained from Eq. (24):

∂2Mx∂x2=Nx∂2χz∂x2−∂2Txxz∂x2−Pv(29)

On the other hand, based on (15), the following can be calculated:

Mx=Γ2Nx∂2χz∂x2−Γ2∂2Txxz∂x2−Γ2Pv+Azzεz+Azgεg−e312κ33h312εz−q312κ33AMzgεg(30)

The following equation is derived by transforming Eqs. (24) and (30):

Nx∂2χz∂x2−Γ2Nx∂4χz∂x4−∂2Txxz∂x2+Γ2∂4Txxz∂x4−Pv+Γ2∂2Pv∂x2−(Azz−q312κ33h312)∂2εz∂x2−(Azg−q312κ33AMzg)∂2εg∂x2=0(31)

The second derivative of Sx is calculated from (25):

∂2Sx∂x2=Nx∂2χz∂x2+∂Qxz∂x−∂2Yxxz∂x2−Pv(32)

On the other hand, Eq. (16) gives:

Sx=Γ2Nx∂2χz∂x2+Γ2∂Qxz∂x−Γ2∂2Yxxz∂x2−Γ2Pv+Agzεz+Aggεg−q312κ33AMzgεz−q312κ33ASggεg(33)

Finally, Eq. (25) is rewritten in the following form:

Nx∂2χz∂x2−Γ2Nx∂4χz∂x4−∂2Yxxz∂x2+Γ2∂4Yxxz∂x4+∂Qxz∂x−Γ2∂3Qxz∂x3−Pv+Γ2∂2Pv∂x2−(Agz∂2εz∂x2+Agg∂2εg∂x2−q312κ33AMzg∂2εz∂x2−q312κ33ASgg∂2εg∂x2)=0(34)

Therefore, the system of equilibrium equations for the beam has the following form:

Nx∂2(χb+χs)∂x2−Γ2Nx∂4(χb+χs)∂x4−∂2Txxz∂x2+Γ2∂4Txxz∂x4+(Azz−q312κ33h312)∂4χb∂x4+(Azg−q312κ33AMzg)∂4χs∂x4−Pv+Γ2∂2Pv∂x2=0(35)

Nx∂2(χb+χs)∂x2−Γ2Nx∂4(χb+χs)∂x4+As∂2χs∂x2−∂2Yxxz∂x2+Γ2∂4Yxxz∂x4+(Agz−q312κ33AMzg)∂4χb∂x4+(Agg−q312κ33ASgg)∂4χs∂x4−Pv+Γ2∂2Pv∂x2=0(36)

The above system of equations is written more specifically as follows:

(A02L∫0L(∂χz∂x)2dx+q31k14hκ33L∫0L∂2χb∂x2dx+q31k14κ33LANg∫0L∂2χs∂x2dx+A0εTh)∂2(χb+χs)∂x2−Γ2(A02L∫0L(∂χz∂x)2dx+q31k14(h−hg)κ33L∫0L∂2χb∂x2dx+q31k14κ33LANg∫0L∂2χs∂x2dx+A0εTh)∂4(χb+χs)∂x4+ATb∂4χb∂x4+ATs∂4χs∂x4−Γ2(ATb∂6χb∂x6+ATs∂6χs∂x6)+(Azz−q312κ33h312)∂4χb∂x4+(Azg−e312κ33AMzg)∂4χs∂x4−Pv+Γ2∂2Pv∂x2=0(37)

(A02L∫0L(∂χz∂x)2dx+q31k14hκ33L∫0L∂2χb∂x2dx+q31k14κ33LANg∫0L∂2χs∂x2dx+A0εTh)∂2(χb+χs)∂x2−Γ2(A02L∫0L(∂χz∂x)2dx+q31k14hκ33L∫0L∂2χb∂x2dx+q31k14κ33LANg∫0L∂2χs∂x2dx+A0εTh)∂4(χb+χs)∂x4+AYb∂4χb∂x4+AYs∂4χs∂x4−Γ2(AYb∂6χb∂x6+AYs∂6χs∂x6)+As∂2χs∂x2+(Agz−q312κ33AMzg)∂4χb∂x4+(Agg−q312κ33ASgg)∂4χs∂x4−Pv+Γ2∂2Pv∂x2=0(38)

The beam’s equilibrium equation system includes nonlocal coefficients, which represent the influence of the flexomagnetic effect, the influence of temperature, and the influence of large strain. These equations are very complicated and cumbersome because they involve two variables vb and vs.

Assuming the nanobeam is subjected to a single support boundary at both ends and it is simply supported at both ends, this condition requires that the displacements χb, χs and the bending moment to be zero at x = 0 and x = L, use a trigonometric series for the solution form and mechanical load, one obtains:

χb=∑n=1∞X0bsin⁡(nπxL)χs=∑n=1∞X0ssin⁡(nπxL)Pv=∑n=1∞P0vsin⁡(nπxL)(39)

in which X0b, X0s, and P0v are the amplitudes of the respective components. In addition, the beam can also be subjected to other boundary conditions such as clamped-clamped, clamped-free, or clamped-simply supported. For these cases, the boundary constraints at both ends differ from those of the simply supported beam, and the selection of displacement functions becomes more complex.

Taking the difference between (37) and (38) yields:

(AYb−ATb+Agz−q312κ33AMzg−Azz+q312κ33h312)∂4χb∂x4+(AYs−ATs+Agg−q312κ33ASgg−Azg+q312κ33AMzg)∂4χs∂x4+Γ2((ATb−AYb)∂6wb∂x6+(ATs−AYs)∂6vs∂x6)+As∂2χs∂x2=0(40)

Eq. (40) is rewritten as follows:

{[AYb−ATb+Agz−q312κ33AMzg−Azz+q312κ33h312]−Γ2(ATb−AYb)(nπL)2}X0b(nπL)2+{(AYs−ATs+Agg−q312κ33ASgg−Agz+q312κ33AMzg)(nπL)2−Γ2(ATs−AYs)(nπL)4−As}X0s=0(41)

If

X0b+X0s=X0(42)

Systems (40) and (41) are solved as follows:

X0b=YbX0(43)

X0s=YsX0(44)

where the coefficients Yb, Ys are provided in the Appendix A.

The system of Eqs. (37) and (38) becomes:

−∑n=1∞∫0L(A02L∫0L(∂χz∂x)2dx+q31k14hκ33L∫0L∂2χb∂x2dx+q31k14κ33LANg∫0L∂2χs∂x2dx+A0εTh)X0(nπL)2sin⁡(nπxL)dx−∑n=1∞∫0LΓ2(A02L∫0L(∂χz∂x)2dx+q31k14hκ33L∫0L∂2χb∂x2dx+q31k14κ33LANg∫0L∂2χs∂x2dx+A0εTh)X0(nπL)4sin⁡(nπxL)dx(45)

+∑n=1∞∫0L(ATb+Azz−q312κ33h312)YbX0(nπL)4sin⁡(nπxL)dx+∑n=1∞∫0L(ATs+Azg−q312κ33AMzg)YsX0(nπL)4sin⁡(nπxL)dx+∑n=1∞∫0LΓ2(ATbYb+ATsYs)X0(nπL)6sin⁡(nπxL)dx+∑n=1∞∫0L(1+Γ2(nπL)2)P0vsin⁡(nπxL)dx=0

−∑n=1∞∫0L(A02L∫0L(∂χz∂x)2dx+q31k14hκ33L∫0L∂2χb∂x2dx+q31k14κ33LANg∫0L∂2χs∂x2dx+A0εTh)X0(nπL)2sin⁡(nπxL)dx−∑n=1∞∫0LΓ2(A02L∫0L(∂χz∂x)2dx+q31k14hκ33L∫0L∂2χb∂x2dx+q31k14κ33LANg∫0L∂2χs∂x2dx+A0εTh)X0(nπL)4sin⁡(nπxL)dx+∑n=1∞∫0L(AYb+Afz−q312κ33AMzg)YbX0(nπL)4sin⁡(nπxL)dx+∑n=1∞∫0L(AYs+Agg−q312κ33ASgg)YsX0(nπL)4sin⁡(nπxL)dx+∑n=1∞∫0LΓ2(AYbYb+AYsYs)X0(nπL)6sin⁡(nπxL)dx−∑n=1∞∫0LAsYsX0(nπL)2sin⁡(nπxL)dx+∑n=1∞∫0L(1+Γ2(nπL)2)P0vsin⁡(nπxL)dx=0(46)