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Influence of flexoelectric effect on the bending rigidity of a Timoshenko graphene-reinforced nanorod

  • Subhash K. Nevhal ORCID logo , Madhur Gupta ORCID logo and Shailesh I. Kundalwal ORCID logo EMAIL logo
Published/Copyright: August 4, 2023
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Journal of the Mechanical Behavior of Materials
From the journal Journal of the Mechanical Behavior of Materials Volume 32 Issue 1

Abstract

The focus of this work is to study the influence of flexoelectric phenomenon on the electromechanical response of graphene-reinforced nanocomposite (GNC) nanorods. An analytical model has been derived by utilizing the Timoshenko beam theory and the principle of variational work by incorporating flexoelectric effects. The GNC nanorod is subjected to a concentrated load acting downward for clamped-free and simply supported support types. The GNC is reinforced with a defective graphene sheet as it is known to show enhanced polarization. The elastic properties of defective graphene sheets have been evaluated using molecular dynamic simulations. The outcome of our model shows that the flexoelectric effect must be considered for accurate modeling of nanostructures. Irrespective of the support type, flexoelectric effect improves the stiffness of the nanorod. We also observed that the stiffness of the nanorod is significantly influenced by the support type. This work presents an opportunity for the development of high-performance graphene-based nanoactuators/sensors.

Keywords: graphene; flexoelectricity; piezoelectricity; nanocomposite; elastic properties; Timoshenko beam theory; nanorod

1 Introduction

Piezoelectricity is the inherent property of certain dielectric materials to electrically polarize in response to mechanical stimuli. It is well known that piezoelectric crystals are noncentrosymmetric, due to the absence of inversion symmetry. However, the breaking of inversion symmetry induces polarization in noncentrosymmetric dielectrics. This electromechanical coupling is termed as flexoelectric effect, which was identified for the first time by Mashkevich and Tolpygo [1]. Unlike piezoelectric effect, flexoelectric effect exists in all dielectric materials, and it reflects the relationship between strain gradient and polarization [2]. One of the most important and unique properties of flexoelectric effect is size dependency. The effect of flexoelectricity is more pronounced and prominent at a nano-scale. Owing to its unique properties, flexoelectricity has attracted significant attention from the scientific community and has been broadly applied in nano/micro electromechanical systems (NEMS/MEMS).

In this context, Kogan [3] presented a theoretical model to estimate the flexoelectric coefficient of dielectrics. Tahantsev [4] developed a theoretical model to study the flexoelectric response in solid, dielectric crystals and stated that flexoelectric effect differs from piezoelectric effect. Ma and Cross [5] experimentally investigated strain gradient-induced polarization in lead zirconate titanate (PZT) ceramic and observed a higher flexoelectric coefficient in the order of μC/m. Ma and Cross [6] investigated the flexoelectric effect in PZT using a cantilevered beam approach. Based on their experimental investigation, they observed that flexoelectric polarization increases with temperature. Hu and Shen [7] studied the piezoelectric and flexoelectric effects in nano dielectrics using the variational principle. Jiang et al. [8] presented the potential role of flexoelectric sensors and actuators in the application of bio-mechanical systems. Yan and Jiang [9] studied the influence of flexoelectric effect on the electromechanical behavior of nanobeams under various support types. In their work, they reported that flexoelectric effect is more pronounced in nanobeams with smaller thickness. Utilizing the Bernoulli–Euler beam model, Liang et al. [10] studied the electromechanical behavior of piezoelectric nanobeams. They observed that the flexoelectricity and surface effects are size-dependent properties, and their influence is prominent at nanoscale. Zhang and Jiang [11] investigated the effect of flexoelectricity and surface on the static and dynamic response of thin nanoplates using the Ritz approximation solution. They observed that size-dependent effects are more dominant for thinner plates with smaller thickness. Liang et al. [12] developed an analytical model based on Euler–Bernoulli (EB) beam hypothesis to study the effect of flexoelectricity in the nanowire. They reported that due to the effect of flexoelectricity, Young’s modulus and bending rigidity of nanowires show significant improvement. Qi et al. [13] in their work reported that the sharp gradients of the electric and polarization fields emerge close to the surfaces due to the flexoelectric effect. Ray [14] presented an exact solution for the static bending response of nanobeam embedded with a flexoelectric layer acting as an actuator. Wang et al. [15] developed a finite difference method to study the effect of flexoelectricity on the static bending response of piezoelectric nanoplates. Zeng et al. [16] presented an analytical model based on the modified couple stress theory to investigate the nonlinear dynamic analysis of flexoelectric nanoshells. Based on the Kirchhoff plate theory, Wang and Li [17] studied the effect of flexoelectricity on the natural frequency of piezoelectric nanoplates. Yurkov et al. [18] derived a theoretical model using variational principles to study the polarization induced due to flexoelectricity in non-homogeneously heated nanoplates. Su and Zhou [19] utilized the non-local effects of flexoelectricity nanosensors to study the electromechanical response of nanobeam. Beni [20,21] utilized a modified non-classical flexoelectric theory to study the effect of size-dependent properties on the static and free vibration analysis of micro/nanotubes. Most recently, Gupta et al. [22] investigated the flexoelectric response in boron-nitride-based nanocomposite beams for various support types. They observed that bulk flexoelectricity stiffens the nanobeam for all support types. The aforementioned studies suggest that at nanoscale the flexoelectric effect should be taken into account for the accuracy of the model.

Graphene, a one-atomic, thick, 2D planner sheet was first discovered by Novoselov et al. in 2004 [23]. Owing to its unique structure and exceptional thermo-electro-mechanical properties, graphene garnered immense attention from researchers [24,25,26]. Andrew et al. [27] presented a theoretical model to calculate the mechanical properties of graphene. They observed that graphene demonstrated excellent resilience to stretching with a value of 206.6 N / m . Lee et al. [28] carried out an experimental investigation on reduced graphene oxide decorated with dopamine. They observed that due to the incorporation of dopamine, the mechanical properties of graphene oxide improved, i.e., it showed better thermal stability and enhanced electrical conductivity. Liu et al. [29] in their work discussed the naturally present defects in graphene sheets and the effects of these defects on the material property of graphene. Kundalwal et al. [30] in their findings showed the presence of strain gradient in the non-piezoelectric graphene sheets. They also observed strong polarization in graphene sheets due to irregular redistribution of the electron density. Most recently, Nevhal and Kundalwal [31] investigated the effects of various defects on the polarization in graphene nanoribbons using density functional theory.

With the recent advancement, graphene is extensively used as structural reinforcement in polymer composites. It was observed that incorporating graphene as reinforcement results in a superior composite structure. Zhao et al. [32] developed graphene-based polymer composites and observed a 150% improvement in tensile strength at low loading, wherein Young’s modulus increased by ∼ 10 times. Tang et al. [33] studied the effect of graphene dispersion on the mechanical properties of graphene reinforcement composites. Their outcomes revealed that high dispersion of graphene results in a significant enhancement in electrical conductivity and fracture toughness of the composite. Due to the presence of strong polarization, graphene-based piezoelectric composites have multifarious NEMS/MEMS applications. Ray et al. [34,35,36] developed various micromechanical and homogenization models to evaluate the elastic and piezoelectric properties of 1–3 PZCs. They found that such PZCs showed an enhanced out-of-plane piezoelectric coefficient. Gupta et al. [37,38] extensively used such PZCs to actively control the dynamic vibrations of smart laminated beams and plates using active control layer damping treatment. Based on the nonlocal elastic theory, Li et al. [39] investigated the nonlinear vibrations of graphene/piezoelectric sandwich composite. They observed that as the size effect increases, the nonlinear resonant frequency decreases. Making use of first-order shear deformation theory, Song et al. [40] performed a dynamic analysis on functionally graded multilayer graphene nanoplatelet (GPL)/polymer composite plates for forced and free vibrations. Feng et al. [41] studied the nonlinear bending analysis of nanocomposite beams reinforced with non-homogeneously distributed GPLs. They observed substantial improvement in the bending performance of nanobeams due to the incorporation of small amount of GPLs. Justino et al. [42] utilized graphene to fabricate sensors and biosensors due to its electromechanical properties. Kundalwal et al. [43,44] developed an analytical model to investigate the electromechanical response of graphene/polymer composite nanowires accounting for the effects of flexoelectricity. They observed that at nanoscale the effects of flexoelectricity on the electromechanical response of nanowires were noteworthy. Using EB beam model, Chen et al. [45] studied the dynamic response of graphene-reinforced porous nanocomposite beams considering the flexoelectric effects. Their outcomes reveal that porosity and flexoelectricity can significantly affect the vibrational behavior of nanobeams.

The existing work on graphene-based composite structures shows that graphene has a potential application as a structural reinforcement due to its exceptional elastic and electrical properties. However, a few studies are available in the literature that focus on the size-dependent response of graphene-reinforced nanocomposite (GNC). To the best of the authors’ knowledge, no study is available on the bending and rotational response of GNC by utilizing the Timoshenko beam theory. The present work is organized into various sections. Section 2 deals with the theoretical formulation of Timoshenko beam model by incorporating the effects of flexoelectricity. Section 3 deals with the estimation of the elastic properties of defective graphene sheets using molecular dynamics (MD) simulations. The outcomes of the present work are shown in Section 4. The effect of flexoelectric phenomenon on the bending rigidity and cross-sectional rotation of GNC nanorods has been adequately studied in this work for clamped-free (CF) and simply supported (SS) support types. Outcomes reveal that flexoelectricity significantly affects the flexural rigidity of the nanorod and that at nanoscale its effects must be taken into account for accurate modeling of nanostructures.

2 Electromechanical behavior of GNC nanorod

In this section, a Timoshenko beam model is utilized to derive the governing equations for the GNC nanorod by considering the effects of flexoelectricity for CF and SS support types. For the bulk piezoelectric nanostructure, the electric Gibbs free energy density function U b can be written as follows [46]:

(1) U b = − 1 2 a kl E k E l + 1 2 c ij kl ε ij ε kl − e ijk ε ij E i − f ijkl E i η jkl + r ijklm ε ij η klm + 1 2 g ijklmn η ijk η lmn .

For the sake of simplicity, the last two terms appearing in Eq. (1) are neglected in this work which are of fifth-order tensor. Such assumption was made by Majdoub et al. [47] in their work and was validated by MD simulations. The benefit of such consideration is that continuum piezoelectricity models considering the flexoelectricity can be used to study the nanoscale piezoelectricity in a computationally expedient manner rather than using atomistic calculations which have clear computational limits in terms of system size and computational expense. Here, a, c, e, f, r, and g are the material property tensors. Specifically, the dielectric constant, elastic constants, classical piezoelectric constant, and flexoelectricity constant are represented by a, c, e, and f, respectively. It should be noted that tensors e and f represent the electromechanical coupling and will be equal to zero if the electromechanical coupling is not taken into account. The strain and strain gradient components are given as follows [48]:

(2) ε ij = 1 2 ( μ i , j + μ j , i ) ,

(3) η jkl = ε jk , l = 1 2 ( μ j , kl + μ k , jl ) .

The electric field can be written as follows:

(4) E i = − φ i .

The constitutive equations for bulk piezoelectric material derived from the internal energy density can be expressed as follows:

(5) σ ij = ∂ U b ∂ ε ij = c ijkl ε kl − e kij E k τ jkl = ∂ U b ∂ η jkl = − f ijkl E i D k = − ∂ U b ∂ E k = a kl E l + e kij ε ij + f klij η ijl ,

where σ ij , D k , and τ jkl represent the Cauchy stress tensor, electric displacement vector, and higher-order stress, respectively. It is noted that σ ij = σ ji , τ ijm = τ jim .

Figure 1 Schematics of nanorod under concentrated point load (a) CF and (b) SS boundary condition.
Figure 1

Schematics of nanorod under concentrated point load (a) CF and (b) SS boundary condition.

In the present work, a Cartesian coordinate system (x, y, z) is used to describe the nanorod as shown in Figure 1. The neutral axis of the nanorod is taken along the x-axis, whereas the thickness of the rod is taken along the z-axis. As shown in Figure 1, a point load F is applied at the free end of CF nanorod (x = l) and at the center of the SS nanorod (x = l/2), respectively. As per the classical Timoshenko beam theory, the displacement field equations can be expressed as follows [49]:

u x ( x , y , z ) = − z ϕ ( x ) ,

u y ( x , y , z ) = 0 ,

(6) u z ( x , y , z ) = w ( x ) ,

where transverse displacement is shown by w ( x ) and cross-section rotation is given by ϕ ( x ) . The non-zero strains and strain gradients can then be obtained from Eqs. (2) and (6) as follows:

(7) ε 11 = − z d ϕ d x , ε 13 = ε 31 = 1 2 − ϕ + d w d x ,

η 111 = ∂ ε 11 ∂ x = − z d 2 ϕ d x 2 ,

η 113 = ∂ ε 11 ∂ z = − d ϕ d x ,

(8) η 131 = η 311 = 1 2 − d ϕ d x + d 2 w d x 2 .

The material property matrices are as follows:

c ijkl = c 11 c 12 c 13 c 12 c 11 c 13 c 13 c 13 c 33 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 44 0 0 0 c 44 0 0 0 c 66 ,

(9) a kl = a 11 0 0 0 a 11 0 0 0 a 33 ,

e kij = 0 0 0 0 0 0 e 31 e 31 e 33 0 e 15 0 e 15 0 0 0 0 0 ,

where the Voigt notation are considered as 11 → 1 , 22 → 2 , 33 → 3 , 23 → 4 , 13 → 5 , 12 → 6 .

For the flexoelectric coefficients, Quang and He [50] provide the various rotational symmetries for flexoelectric tensors. For the crystalline medium, the possible symmetry of flexoelectric coefficient is discussed by Shu et al. [46]. In the present work, we considered the flexoelectric coefficients as follows [46,51]:

f 1111 = f 2222 = f 3333 = f 11 ,

(10) f 1313 = f 2323 = f 1212 = f 2112 = f 3223 = f 3113 = f 111 ,

f 1122 = f 1133 = f 2211 = f 2233 = f 3322 = f 3311 = f 14 .

In case of rods, the thickness of the rod is smaller than its length; thus it can be assumed that the electric field exists only in the thickness direction, i.e., E 1 = E 2 = 0 , E 3 ≠ 0 . In case of open circuit condition, the electric displacement on the surface should be zero ( D 3 | s = 0 ) . In the absence of free electric charge, electric displacement satisfies Gauss’s law and is given by ∂ D 3 / ∂ z = 0 . Thus, the internal electric field can be given as follows:

(11) E 3 = − e 31 a 33 ε 11 − f 111 a 33 η 113 − 2 f 14 a 33 η 131 .

From Eqs. (5), (7), (8), and (11) the non-zero stresses and non-zero moment stresses can be given as follows:

σ 11 = c 11 + e 31 2 a 33 ε 11 + e 31 f 111 a 33 η 113 + 2 e 31 f 14 a 33 η 131 ,

σ 13 = 2 k c 44 ε 13 ,

τ 113 = e 31 f 14 a 33 ε 11 + f 111 f 14 a 33 η 113 + 2 f 14 2 a 33 η 131 ,

(12) τ 131 = τ 311 = e 31 f 111 a 33 ε 11 + f 111 2 a 33 η 113 + 2 f 111 f 14 a 33 η 131 .

Here k is the shear correction factor, and for the present work we consider k = 1. The electric Gibbs free energy can be expressed as follows:

δ ∫ V U b d V = ∫ V ∂ U b ∂ ε ij ∂ ε ij + ∂ U b ∂ η jkl δ η jkl − ∂ U b ∂ E k δ E k d V

= ∫ V ( σ ij δ ε ij + τ jkl δ η jkl − D k δ E k ) d V = ∫ V ( σ 11 δ ε 11 + 2 σ 13 δ ε 13 + τ 113 δ η 113 + 2 τ 131 δ η 131 ) d V

= ∫ 0 l − ( M + P ) d δ ϕ d x + Q − δ ϕ + d δ w d x + R − d δ ϕ d x + d 2 δ w d x 2 d x

(13) = ∫ 0 l d ( M + P ) d x − Q − d R d x δ ϕ d x + ∫ 0 l d d x Q − d R d x δ w d x − [ ( M + P ) + R ] δ ϕ | 0 l + R d δ w d x 0 l − Q − d R d x δ w w | 0 , l

where the resultant shear force and the resultant bending moment are given by the following:

Q = ∫ A σ 13 d A = c 44 A − ϕ + d w d x ,

M = ∫ A σ 11 z d A = − c 11 + e 31 2 a 33 I d ϕ d x ,

P = ∫ A τ 113 d A = − f 14 f 111 a 33 A d ϕ d x + f 14 2 a 33 A − d ϕ d x + d 2 w d x 2 ,

(14) R = ∫ A τ 131 d A = − f 111 2 a 33 A d ϕ d x + f 14 f 111 a 33 A − d ϕ d x + d 2 w d x 2 .

The total energy of the overall system is given by Π = ∫ V ( σ ij ε ij + τ ijk η ijk ) d V − ∫ a ( p ̅ i u i + r ̅ i Δ u i ) d a , in which W = ∫ a ( p ̅ i u i + r ̅ i Δ u i ) d a is the work done by the external force. For the CF support condition, the work is given by W = F w | x = l and for the SS support condition, the work is given by W = F w | x = l / 2 . Making use of the variational principle, δ Π = 0 , the governing equations can be expressed as follows:

(15) d d x ( M + P ) = Q − d R d x , d d x Q − d R d x = 0 ,

and the corresponding boundary conditions prescribed at the end of the nanorod (x = 0 and x = l) are as follows:

(16) M + P + R or ϕ Q − d R d x or w R or d w d x .

The boundary conditions for CF nanorod can be given by the following:

(17) ϕ | x = 0 , ( M + P + R ) | x = l = 0 , w | x = 0 = 0 , Q − d R d x = − F , d w d x x = 0 = 0 , R | x = l = 0 .

For SS nanorod, the necessary boundary conditions are given by the following:

(18) ( M + P − R ) | x = 0 = 0 , ϕ | x = l 2 = 0 , w | x = 0 = 0 , Q − d R d x x = l 2 = − F 2 , R | x = 0 = 0 , d w d x x = l 2 = 0 .

Substituting Eq. (14) into Eq. (15), the governing equations accounting for the flexoelectric effect can be expressed as follows:

(19) c 11 + e 31 2 a 33 I + ( f 14 + f 111 ) 2 a 33 A d 2 ϕ d x 2 − f 14 ( f 14 + f 111 ) a 33 A d 3 w d x 3 + c 44 A − ϕ + d w d x = 0 c 44 A − d ϕ d x + d 2 w d x 2 + f 111 a 33 A d 3 ϕ d x 3 − f 14 f 111 a 33 A − d 3 ϕ d x 3 + d 4 w d x 4 = 0 .

For the cross-section rotation, the solutions for Eq. (19) can be obtained by considering the following [52]:

(20) θ ( x ) = − ϕ ( x ) + d w ( x ) d x ,

where θ ( x ) indicates twice the rigid body rotation of the beam element. Substituting the derivation of Eq. (20) in Eq. (19), one can obtain the following relation:

(21) c 11 + e 31 2 a 33 I + f 14 f 111 a 33 A d 3 ϕ d x 3 + f 14 2 a 33 A d 3 θ d x 3 = 0 .

Substituting Eq. (21) into Eq. (19) yields the following:

(22) B d 3 θ d x 3 − C d θ d x = 0 ,

where B = f 14 f 111 a 33 1 − f 14 f 111 A ( c 11 a 33 + e 31 2 ) I + f 14 f 111 A and C = c 44 A . It is noted that B > 0 for 0 < f 14 f 111 A < ( c 11 a 33 + e 31 2 ) I + f 14 f 111 A and C > 0 for c 44 > 0 .

The general solution for Eq. (22) can then be obtained as follows:

(23) θ ( x ) = C 4 cosh ( kx ) + C 5 sinh ( kx ) − t 2 t 3 t 1 C 1 .

The terms appearing in Eq. (23) are given by the following:

k 2 = c 44 A t 2 t 1 , t 1 = c 11 + e 31 2 a 33 I f 14 f 111 A a 33 , t 2 = c 11 + e 31 2 a 33 I + f 14 f 111 A a 33

and

t 3 = c 11 + e 31 2 a 33 I + f 14 2 A a 33 t 2 .

where C 4 , C 5 , and C 1 are the undetermined parameters.

Substituting Eq. (23) into (21), the cross-section rotation of nanorod can be expressed as follows:

(24) ϕ = − f 14 2 A a 33 t 2 C 4 cosh ( kx ) + C 5 sinh ( kx ) − t 2 t 3 t 1 C 1 + 1 2 C 1 x 2 + C 2 x + C 3 ,

where C i ( i = 1 − 5 ) are unknown parameters that can be predicted using the boundary conditions.

Substituting Eqs. (23) and (24) into Eq. (20), the analytical solution for the transverse deflection can be given by the following:

(25) w ( x ) = − t 3 t 2 1 k C 4 sinh ( kx ) + 1 k C 5 cosh ( kx ) + t 2 t 3 t 1 k 2 + 1 6 C 1 x 3 + 1 2 C 2 x 2 + C 3 x + C 6 .

Invoking the boundary conditions given by Eqs. (17) and (18), the unknown constants C i ( i = 1 – 5 ) can be calculated. For CF nanorod, the unknown constants are given by the following:

(26) C 1 = − F t 3 + f 14 f 111 A a 33 , C 2 = − C 1 l , C 3 = 0 , C 4 = F c 44 A f 14 f 111 a 33 t 3 − 1 , C 5 = − sinh ( kl ) cosh ( kl ) C 4 , C 6 = t 3 t 2 k C 5 .

Unknown constants for SS nanorod are given by the following:

(27) C 1 = F 2 t 2 , C 3 = − c 1 l 2 8 , C 4 = − t 2 t 3 C 1 t 1 k 2 cosh kl 2 , C 2 = C 5 = C 6 = 0 .

The governing Eq. (19) can be reformulated as follows when the flexoelectric effect is ignored:

(28) c 11 + e 31 2 a 33 I d 2 ϕ d x 2 + k c 44 A − ϕ + d w d x = 0 k c 44 − d ϕ d x + d 2 w d x 2 = 0 .

The governing equation for the classical Timoshenko beam theory can be obtained by further ignoring the piezoelectric effect [53].

3 Mechanical properties of defective graphene

The literature suggests that polarization of defective graphene sheet increases due to the breaking of inversion symmetry and the presence of strain gradients [30,31]. However, the elastic properties of such defective graphene sheets are not available in the literature. Consequently, they need to be predicted for further studies. Therefore, we estimated the elastic properties of defective graphene sheets with 6.43% triangular defect using MD simulations. Figure 2 shows the schematics of such a defective graphene sheet. For the purpose of this work, a large-scale atomic/molecular massively parallel simulator [54] has been utilized to perform the MD simulations. The molecular interactions between the carbon–carbon (C–C) atoms of the graphene sheet are described with the adaptive intermolecular reactive empirical bond order force fields [55]. The atomic volume of the relaxed defective graphene sheet is calculated with a thickness of 3.4 Å [56,57]. Overall stress developed in the defective graphene sheet was evaluated by averaging the stress developed on each carbon atom. Later, tensile loading is applied to predict Young’s modulus (E) and Poisson’s ratio by plotting stress–strain curves. The detailed procedure of MD simulations is provided by Kundalwal and Choyal [58]. The material properties of the defective graphene sheet are shown in Table 1.

Figure 2 Armchair graphene sheet with trapezoidal pore subjected to axial stress with 6.43% vacancies.
Figure 2

Armchair graphene sheet with trapezoidal pore subjected to axial stress with 6.43% vacancies.

Table 1

Material properties of graphene and polyimide matrix

Material E (GPa) μ e 31 (C/m2) e 33 (C/m2) a 33 (F/m)
Pristine graphene 985 (present) 0.265 (present) −0.221 [30] 0.221 [30] 1.106 × 10−10 [63]
Graphene with 4.5% vacancy 969 (present) 0.265 (present) −0.027 [30] 0.027 [30] 1.106 × 10−10 [63]
Graphene with 6.43% vacancy 960 (present) 0.265 (present) −0.051 [31] 0.051 [31] 1.106 × 10−10 [63]
Polyimide 4.2 [64] 0.4 [64] − − 3.009 × 10−11 [65]

The results obtained for the pristine graphene sheet are validated and are found to be in agreement with the literature regarding various modeling techniques and experimental investigations [59,60,61]. The results of the defective graphene sheets with 4.5% vacancies are validated with those reported by Jing et al. [60] and are found to be in agreement with the literature. It can be observed from Table 1 that Young’s modulus of the graphene sheet was not much affected because of the defects. This is attributed to the hydrogenation and saturation of the dangling bonds at the edges and to the porosity of the graphene sheet.

4 Results and discussion

This section deals with the investigation of the effects of flexoelectricity on the electromechanical behavior of GNC nanorods using the Timoshenko piezoelectric beam model. The piezoelectric coefficient of the defective armchair graphene sheet was taken from Nevhal and Kundalwal [31,62] and had 160 atoms, while the length and width of the graphene sheet were 19.88 and 16.0 Å, respectively. The material properties of graphene and polyimide matrix are presented in Table 1. The modeling parameters of the nanowire are taken as follows: h = 50 nm, b = 2 h, and l = 8 h nm, where b, h, and l are the width, thickness, and length of the GNC nanowire, respectively. The point load F = 1 nN is applied at x = l for the CF boundary condition and at x = l / 2 for SS boundary condition.

In the present work, we considered a graphene sheet with 6.43% vacancy defect to study the influence of flexoelectric effect on the electromechanical behavior of GNC nanorod. GNC is composed of the graphene sheet and the polyimide matrix, with graphene fiber reinforced along the 3-axis. Such piezoelectric composite can be termed as 1–3 GNC. To simplify the work, the graphene can be considered as a continuum plate to calculate its bulk properties [66,67]. Many existing studies on the straining of nanomaterials are based on analytical as well as numerical modeling based on the concept of continuum elasticity. The displacement of each atom is given by the deformation of the continuum medium, in which the atom is embedded, for a uniformly deformed GNC. Thus, for the present analysis, the GNC can be used as a continuum medium [68,69,70]. The 1–3 piezoelectric composites have better out-of-plane actuation because of the improved piezoelectric coefficient, e 33 . This is attributed to the alignment of fibers in the 3-direction. Such 1–3 PZCs have been extensively studied by Gupta et al. [37,38,71]. The effective elastic, piezoelectric, and dielectric properties of GNC have been evaluated by a micromechanical model based on the Mori–Tanka approach presented in previous studies [36,72] and tabulated in Table 2.

Table 2

Effective properties of GNC (v g = 0.3)

Material C 11 (GPa) C 12 (GPa) C 44 (GPa) e 31 (C/m2) e 33 (C/m2) a 33 (F/m)
Graphene with 6.43% vacancy 13.8876 8.705 4.014 −0.00182 0.175 6.921 × 10−11

To verify the accuracy of the present model, the normalized deflections of the nanobeam are verified with the available results of the EB beam model presented by Gupta et al. [22] for the identical nanobeam. Table 3 shows the comparison of normalized deflections by considering the effects of flexoelectricity. Table 3 shows good agreement between the present Timoshenko beam theory and EB beam theory. It can be seen from Table 3 that the EB beam model underpredicts the deformation of nanobeam. This is attributed to the fact that the EB beam theory considers that the cross-section of the beam is always perpendicular to the neutral axis after deformation. The Timoshenko beam theory is the superior version of the EB beam theory as it accounts for the deformation due to shear by considering the shear lag correction factor.

Table 3

Normalized deflection of nanobeam with various boundary conditions [22]

Thickness Boundary condition x/L Model Normalized deflection
20 mm CF 0.2 Present −2.84
EB −2.72
0.6 Present −19.36
EB −18.53
1.0 Present −40.62
EB −38.98
20 mm SS 0.1 Present −1.20
EB −1.15
0.3 Present −3.01
EB −2.99
0.5 Present −3.82
EB −3.68

The results presented in this work demonstrate that the deflection predicted by the current model (Timoshenko beam theory considering the effect of flexoelectricity) is smaller than that of the classical Timoshenko beam theory. This disparity highlights the influence of flexoelectricity on the effective bending rigidity of the nanobeam. With the inclusion of flexoelectric effect, the beam exhibits a significantly higher bending rigidity compared to conventional beams, leading to a stiffer bending behavior under purely mechanical loads. Additionally, the rotational displacement of the current Timoshenko beam’s cross-section was also found to be smaller than the prediction of the classical Timoshenko beam theory.

Figure 3 illustrates the deflection of the cantilevered nanorod with the aspect ratio. The maximum deflection for the CF nanorod is noted at its free end i.e., x = l. It can be observed from the figure that the GNC nanorod with flexoelectric effect shows smaller deflections as compared to the classical Timoshenko nanorod. The maximum deflection of nanorod (x = l) is reduced by ∼ 50 % due to the consideration of flexoelectric effects. This is mainly because flexoelectric effects improve the effective bending rigidity of the GNC nanorod. Thus, owing to flexoelectric effects, the GNC nanorod shows stiffer behavior under mechanical loading as compared to the classical nanorod. Figure 4 demonstrates the cross-section rotation of the nanorod with respect to the aspect ratio. The figure depicts that due to flexoelectric effects, the flexural rigidity of the nanorod improves, which results in smaller cross-section rotation compared to the classical nanorod. The maximum cross-section rotation is observed at the free end of the nanorod. Based on our observation, we say that flexoelectricity can be used to control the displacement profile of a piezoelectric nanorod at the nanoscale, which is useful for the design of piezoelectric nanorod-based actuators.

Figure 3 Deflection of CF piezoelectric nanorod with respect to its aspect ratio.
Figure 3

Deflection of CF piezoelectric nanorod with respect to its aspect ratio.

Figure 4 Cross-sectional rotation of CF nanorod with respect to its aspect ratio.
Figure 4

Cross-sectional rotation of CF nanorod with respect to its aspect ratio.

Figures 5 and 6 show the deflection and rotation of the SS nanorod with its aspect ratio. For the SS boundary condition, the deflection and rotation of the nanorod are shown for one-half of its length by taking advantage of symmetry. A concentrated load of 1 nN is applied at the center x = l / 2 of the nanorod. As expected, the deflection of the nanorod decreases due to the incorporation of flexoelectric effects. As compared to the classical GNC nanorod, the nanorod with flexoelectric effects shows ∼44% decrement in the maximum static deflections. For the nanorod, the maximum deflection is observed at its center and it becomes zero at its supports, i.e., at x = 0 and x = l , as shown in Figure 5.

Figure 5 Deflection of SS piezoelectric nanorod with respect to its aspect ratio.
Figure 5

Deflection of SS piezoelectric nanorod with respect to its aspect ratio.

Figure 6 Cross-sectional rotation of SS nanorod with respect to its aspect ratio.
Figure 6

Cross-sectional rotation of SS nanorod with respect to its aspect ratio.

The cross-section rotation of GNC nanorod with or without considering flexoelectric effects for the SS boundary condition is shown in Figure 6. The symmetry of this figure lies in the fourth quadrant. As can be seen, the rotation of the GNC nanorod, considering the effects of flexoelectric effects, is less compared to the classical nanorod. For the case of SS boundary condition, the maximum rotation is observed at the supports, while the cross-section of the nanorod remains unchanged at the center.

From Figures 3–6, it can be observed that flexoelectricity greatly influences the flexural rigidity of the structure. Flexoelectric effect stiffens the nanorod for both CF and SS support conditions. However, for the same loading condition, the CF nanorod shows softer behavior, while the SS nanorod shows stiffer behavior. This is attributed to the fact that the curvature of the CF nanorod is concave downward with a negative slope. Hence, the CF nanorod shows larger deflections when mechanical load is applied. In contrast, the curvature of the SS nanorod is concave upward with a positive slope that opposes the applied load. Thus, the SS nanorod shows smaller deflection and stiffer behavior compared to the CF nanorod. Hence, it can be inferred from this discussion that flexoelectricity is crucial and we cannot ignore the flexoelectric effect at the nanoscale level which may lead to inaccurate results. This size-dependent phenomenon can be utilized in the design and fabrication of pressure and force-based nano sensors/actuators.

5 Conclusion

In the present work, an analytical model has been derived by utilizing the Timoshenko beam theory and the principle of variational work. Based on the present model, the effect of flexoelectricity on the deflection and cross-section rotation of GNC nanorods is investigated by considering the CF and SS support types. Our results reveal that size-dependent properties like flexoelectricity greatly influence the flexural rigidity of the GNC nanorod. For both the support conditions, flexoelectric effect is found to enhance the stiffness of the nanorod, which results in smaller deflections and cross-sectional rotations of the nanorod than that of the classical nanorod. Furthermore, it was observed that for the same mechanical loading, the SS GNC nanorod showed stiffer behavior, while the CF GNC nanorod showed softer behavior due to curvature effects. In conclusion, our fundamental outcomes indicate that at nanoscale flexoelectric effects must be taken into account for accurate modeling of nanostructures subject to mechanical loading.

Acknowledgments

This work was generously supported by the Science Engineering Research Board (SERB), Department of Science and Technology, Government of India. The corresponding author (S.I.K.) acknowledges the financial support of the SERB Core Research Grant (CRG/2022/000786) awarded to him.

  1. Author contributions: S.K. Nevhal: data curation, formal analysis, investigation, methodology, software, validation, visualization, writing – original draft. M. Gupta: formal analysis, investigation, software, validation, visualization, writing – review and editing. S.I. Kundalwal: conceptualization, funding acquisition, investigation, methodology, project administration, resources, supervision, writing – review and editing.

  2. Conflict of interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.

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Received: 2022-12-22
Revised: 2023-07-02
Accepted: 2023-07-09
Published Online: 2023-08-04

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/jmbm-2022-0295
Keywords for this article
graphene; flexoelectricity; piezoelectricity; nanocomposite; elastic properties; Timoshenko beam theory; nanorod
Creative Commons
BY 4.0

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  34. Investigation of the ability of steel plate shear walls against designed cyclic loadings: Benchmarking and parametric study
  35. Effect of truck and train loading on permanent deformation and fatigue cracking behavior of asphalt concrete in flexible pavement highway and asphaltic overlayment track
  36. The impact of zirconia nanoparticles on the mechanical characteristics of 7075 aluminum alloy
  37. Investigation of the performance of integrated intelligent models to predict the roughness of Ti6Al4V end-milled surface with uncoated cutting tool
  38. Low-temperature relaxation of various samarium phosphate glasses
  39. Disposal of demolished waste as partial fine aggregate replacement in roller-compacted concrete
  40. Review Articles
  41. Assessment of eggshell-based material as a green-composite filler: Project milestones and future potential as an engineering material
  42. Effect of post-processing treatments on mechanical performance of cold spray coating – an overview
  43. Internal curing of ultra-high-performance concrete: A comprehensive overview
  44. Special Issue: Sustainability and Development in Civil Engineering - Part II
  45. Behavior of circular skirted footing on gypseous soil subjected to water infiltration
  46. Numerical analysis of slopes treated by nano-materials
  47. Soil–water characteristic curve of unsaturated collapsible soils
  48. A new sand raining technique to reconstitute large sand specimens
  49. Groundwater flow modeling and hydraulic assessment of Al-Ruhbah region, Iraq
  50. Proposing an inflatable rubber dam on the Tidal Shatt Al-Arab River, Southern Iraq
  51. Sustainable high-strength lightweight concrete with pumice stone and sugar molasses
  52. Transient response and performance of prestressed concrete deep T-beams with large web openings under impact loading
  53. Shear transfer strength estimation of concrete elements using generalized artificial neural network models
  54. Simulation and assessment of water supply network for specified districts at Najaf Governorate
  55. Comparison between cement and chemically improved sandy soil by column models using low-pressure injection laboratory setup
  56. Alteration of physicochemical properties of tap water passing through different intensities of magnetic field
  57. Numerical analysis of reinforced concrete beams subjected to impact loads
  58. The peristaltic flow for Carreau fluid through an elastic channel
  59. Efficiency of CFRP torsional strengthening technique for L-shaped spandrel reinforced concrete beams
  60. Numerical modeling of connected piled raft foundation under seismic loading in layered soils
  61. Predicting the performance of retaining structure under seismic loads by PLAXIS software
  62. Effect of surcharge load location on the behavior of cantilever retaining wall
  63. Shear strength behavior of organic soils treated with fly ash and fly ash-based geopolymer
  64. Dynamic response of a two-story steel structure subjected to earthquake excitation by using deterministic and nondeterministic approaches
  65. Nonlinear-finite-element analysis of reactive powder concrete columns subjected to eccentric compressive load
  66. An experimental study of the effect of lateral static load on cyclic response of pile group in sandy soil
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