Uncertainty analysis of bending response of flexoelectric nanocomposite plate

Madhur Gupta; Pritesh V. Bansod; Heeralal Gargama; Shailesh I. Kundalwal

doi:10.1515/jmbm-2025-0045

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Uncertainty analysis of bending response of flexoelectric nanocomposite plate

Madhur Gupta , Pritesh V. Bansod , Heeralal Gargama and Shailesh I. Kundalwal

Published/Copyright: May 16, 2025

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From the journal Journal of the Mechanical Behavior of Materials Volume 34 Issue 1

Abstract

In this research work, boron nitride nanosheet (BNS)-reinforced nanoplate is analyzed to evaluate its static bending performance including the effect of flexoelectricity and surface subjected to hydrostatic load, point load, and line load conditions. All edges of the nanoplate have simply supported boundary conditions, and it is transversely loaded from the top plane. The Navier method is employed to obtain the solution of the governing equation, and the static bending performance of the nanoplate is exhibited. The impact of uncertainties in the design variables and application loads have also been analyzed to evaluate the performance-based reliability of the bending response. It is noticed that both surface and flexoelectric effects are highly size-dependent and cannot be neglected at the nanoscale. Therefore, these factors are included in the model to investigate their contribution to the static bending of the BNS-reinforced nanoplate.

Keywords: boron nitride-reinforced composite; Navier solution; flexoelectricity; surface effect; nanoplate; performance-based reliability; Monte Carlo simulation

1 Introduction

Recently, 2D materials, such as carbon nanotubes [1,2], graphene [3], and boron nitride nanosheet (BNS) [4,5], are used in polymer matrix composites, and they exhibit excellent thermal and mechanical properties. Hexagonal BNS has attracted researchers’ attention over graphene due to its superiority in physical properties. BNS has a structure similar to the hexagonal structure of graphene, where boron and nitrogen atoms are alternatively placed in a hexagonal pattern. BNS possesses some unique properties compared to graphene, such as a wide bandgap of about 5–6 eV, which turns it into a good electrical insulator, excellent thermal conductivity of 227–280 W/mK, high stability in chemical and high-temperature environments, and high oxidation resistance [6]. Furthermore, the BNS monolayer possesses piezoelectricity as a result of a non-centrosymmetric structure. In the experimental study conducted by Ares et al. [7] using electrostatic force microscopy, it is observed that the monolayer of hexagonal boron nitride nanosheets is piezoelectric. Piezoelectric properties are greatly affected by shape and are almost independent of size [8]. In recent years, Gupta et al. [9,10,11,12] made efforts to develop micromechanical models to study the elastic and piezoelectric properties of 1–3 piezoelectric composites, with piezoelectric fibers reinforced in the thickness direction. Such composites exhibit enhanced out-of-plane piezoelectric constant, which shows better response against the transverse loading. Moreover, properties such as lightweight, high strength, and piezoelectricity make BNS an ideal candidate for nanodevices.

Over the last couple of decades, flexoelectricity effects have received enormous research attention in the development of nano-electro-mechanical-system devices. Flexoelectricity is the impulsive electric polarization generated by the non-uniform strain gradients, which is a function of flexoelectric coefficient and strain gradient magnitude. Flexoelectricity is substantiated in all dielectric materials including centrosymmetric and non-centrosymmetric crystals. Inversion symmetry of the material structure tends to locally discontinue due to non-uniform strain gradients, and as a result, a dipole moment is formed, resulting in polarization [13]. However, flexoelectric effect is extrusive at the nanoscale due to enhanced flexibility and defect-free structures [14]. Apart from the flexoelectric effect, surface phenomena originate from surface elasticity theory [15], which offers a significant contribution to the electromechanical response of the nanoscale structures [16,17]. The size-dependent properties result from the large ratio of volume and surface area of the nanostructures like nanobeams, nanoplates, and nanoshells. These nanostructures are the fundamental building blocks of nanodevices such as sensors, actuators, and energy harvesters. Hence, the combined effects of surface and flexoelectricity should be involved in evaluating the electromechanical responses of nanostructures.

Several studies, which investigate the static bending response of the nanobeams and nanoplates, including surface and flexoelectric effects, are reported in the literature. The electromechanical behavior study of curved nanobeam and nanowire [18,19], sinusoidal piezoelectric nanoplate [20], and piezoelectric nanofilms [21] are predicted using surface effects. The flexoelectric effect of the heterogenous flexoelectric membrane was studied by Mohammadi et al. [22] and predicted the electromechanical response on the membrane. Yang et al. [23] used an analytical formulation derived from the Kirchhoff plate theory to observe the static deflection of flexoelectric nanoplate and concluded that flexoelectricity affects thin composite nanoplate. The unified surface and flexoelectric effect exhibit noteworthy effects on the electromechanical performance of the nanoplates. Specifically, the effect of size is sharply noticed for the plates of smaller thickness [24,25]. Current literature also reported the analysis of bending and vibrational behavior of nanolaminates, nanosandwich structures, and nano bilayer circular plates. The bending and vibration characteristics of nanolaminates [26] are studied under surface and flexoelectric effects using the analytical models for different boundary conditions. They identified that the flexoelectric and surface effects result in stiffer nanolaminates with reduced natural frequency. Kaveh et al. [27] obtained the normalized natural frequencies of a nano sandwich structure made of functionally graded porous core and flexoelectricity-based face sheets on both sides through governing equations based on a new type of power law for porous materials, Hamilton’s principle, and modified couple stress theory. Similarly, a new flexoelectricity-based theoretical model [28] was developed for a piezoelectric layered bilayer circular nanoplate which accounted the influence of surface and flexoelectric effects on radial deflection, central deflection, polarization, and electric potential of the bilayer piezoelectric nanoplate. They noticed that flexoelectricity affects deflection when negative surface residual stress is considered while positive surface residual stress consideration influences deflection behavior controlled by surface effect. Models based on modified classical plate theory are mainly utilized to predict electromechanical response. Recently, Kundalwal and Gupta extensively studied the influence of flexoelectricity and surface on the bending rigidity of graphene/BNS-based nanostructures such as nano beams, plates, and rods. They developed various size-dependent analytical and finite element models based on theories such as Timoshenko beam theory [29], Euler Bernoulli beam theory [30,31], and Kirchhoff plate theory [32]. Yue et al. [33] investigated the static bending and free vibration properties of a variable width piezoelectric nanoplate with flexoelectricity. They derived the governing equations from the nonlinear Mindlin plate theory, flexoelectricity theory, and finite element theory. They showed that the flexoelectric effect can strongly influence the maximum deflection, the morphology of deformation, and the natural frequency of variable-width piezoelectric nanoplate. This concludes that nanostructures exhibit a significant reduction in bending deflection when surface and flexoelectric effects are simultaneously considered. Additionally, there are uncertainties and stochasticity associated with the design variables and parameters due to the inherent variability in material properties during manufacturing, structural dimensions, and applied loading conditions. The bending response of the beam under such uncertainties cannot be precisely evaluated.

Performance-based reliability analysis (PBRA) is the statistical tool employed to evaluate the reliability performance of structures and components under a specified working environment. PBRA focuses primarily on the relationship between the performance of system and the capability to achieve the predefined level of performance. Recently, the implementation of PBRA in the evaluation of fatigue life performance of complex engineering structures has been reported. Yang et al. [34] reviewed the effect of hydrogen environment on the very high cycle fatigue performance of metals. Meng et al. [35] proposed an intelligent optimization-inspired support vector regression modeling methodology to build a high-precision approximation model for fatigue life estimation of complex engineering structures like support structures of offshore wind turbines. Similarly, Liu et al. [36] employed the bootstrap error circle method of random load signals for uncertainty in the fatigue life prediction of parts. Yang et al. [37] reviewed the general formulation of the reliability-based design optimization strategy and most probable point search algorithm and proposed an RBDO strategy for various marine structures. Luo et al. [38] proposed a highly efficient structural reliability analysis method coupling the active Kriging algorithm with the conjugate first-order reliability method which offers robust and accurate performance.

In this study, BNS-reinforced nanoplate, which is simply supported on all edges, and loaded with a hydrostatic, point, and line load is considered for the study. The governing equation of the BNS nanoplate is obtained using classical plate theory, and Navier solution methodology is adopted to derive the analytical expression for the evaluation of static bending deflection of BNS nanoplate. The PBRA [39,40] of the bending response under the geometric parameters, fiber volume fraction, and applied load has been evaluated using the Monte Carlo simulation (MCS) methodology [41,42]. This technique is favored due to its ease of implementation, robustness, and capacity to provide accurate results that remain unaffected by the dimension or intricacy of the problem, although a substantial number of simulation runs are necessary to achieve a reliable estimate. To address this constraint, we employed the response surface approximation to approximate the bending response in terms of design variables to reduce the computational burden. These approximated polynomials are used to compute the bending response at each design point generated via MCS with the Latin hypercube sampling (LHS) approach [42,43] to evaluate the probability of failure from the limit state function.

Regarding the structure of this article, Section 2 presents the theoretical formulation of the modified Kirchhoff plate theory with the modification for the inclusion of the surface and flexoelectric effects. Section 3 presents the governing differential equation for the BNS-reinforced nanoplate. Section 4 presents a PBRA using the MCS approach followed by results and discussions presented in Section 5. The conclusion of this study is given in Section 6.

2 Theoretical formulation of BNS-reinforced nanoplate

In this section, the governing equation for the uniformly loaded BNS-reinforced nanoplate, which is simply supported over all edges is developed. The derivation is based on the classical plate theory and is modified to cover the surface and flexoelectric effects. The cartesian coordinate system is adopted, and the reference plate is set at the middle layer of the BNS-reinforced nanoplate.

Figure 1 displays the BNS-reinforced nanoplate with a layer of surface loaded with point load, line load, and hydrostatic load. Nanoplate has dimensions of Length a, width b, and thickness h. The thickness of the nanoplate is oriented along the z-direction. This nanoplate contains a surface layer of negligible thickness on its upper and lower surfaces.

Figure 1

(a) Schematic representation of the BNS reinforced nanoplate with top and bottom surface layer subjected to (b) point load, (c) line load, and (d) hydrostatic load.

As per the classical plate theory, the generalized lateral displacement of the nanoplate is given as [44]

(1) u ( x , y , z , t ) = u 0 ( x , y , t ) − z ∂ w ( x , y , t ) ∂ x ,

(2) v ( x , y , z , t ) = v 0 ( x , y , t ) − z ∂ w ( x , y , t ) ∂ y ,

(3) w ( x , y , z , t ) = w ( x , y , t ) ,

where u 0 and v 0 represent the in-plane displacement of the midplane along x and y directions, respectively. Lateral displacement of the mid-plane is expressed by w , considered along the thickness direction and represents the deformation of the mid-plane. At the middle plane, corresponding components of nonzero strain can be expressed as

(4) ε xx = − z ∂ 2 w ∂ x 2 , ε yy = − z ∂ 2 w ∂ y 2 , ε xy = − z ∂ 2 w ∂ x ∂ y .

Kirchhoff plate theory assumes a very small thickness as compared with its in-plane dimensions. Therefore, strain gradient components in the in-plane directions can be neglected when compared with strain gradients in the thickness direction. Past research studies [24,45] have also mentioned similar assumptions for the Euler beam model. Hence, a non-zero strain gradient in the thickness direction is written as

(5) ε xx , z = − ∂ 2 w ∂ x 2 , ε yy , z = − ∂ 2 w ∂ y 2 , ε xy , z = − ∂ 2 w ∂ x ∂ y .

Some previous studies consider the surface effect on strain and stress field [15] and the effect of surface and flexoelectricity on dielectric nanomaterials [46]. Hence, to involve surface and flexoelectric effect, the electric Gibbs free energy is split into a bulk volume and a surface layer of negligible thickness. The bulk electric Gibbs free energy is given by

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

For the piezoelectric material, the last two terms of Eq. (6), r ijklm and g ijklmn , can be excluded [47] and accordingly, the constitutive equation for the bulk dielectric material is written as

(7) σ ij = ∂ U b ∂ ε ij = C ijkl ε kl − e ijk E k ,

(8) τ ijm = ∂ U b ∂ η ijm = − f kijm E k ,

(9) D i = − ∂ U b ∂ E i = e jki ε jk + ∈ ij E j + f ijkl η jkl ,

where ∈ ij is the dielectric tensor, C ijkl is the fourth-order elastic tensor, e ijk is the third-order piezoelectric tensor, f ijkl is the fourth-order tensor of coupling between strain gradient and electric field, σ ij represents the stress tensor, D i is the tensor for the electric field, and E i stands for the vector of the electric field. The strain gradient tensor is given by η jkl and is defined as

(10) η jkl = ε jk , l .

Utilizing Eqs. (7)–(9), the bulk electric Gibbs free energy may be rewritten as

(11) U b = 1 2 σ ij ε ij + 1 2 τ ijm η jkl − 1 2 D k E k ,

and the constitutive relations can be re-written as

(12) σ xx = C 11 ε xx + C 12 ε yy − e 31 E z ,

(13) σ yy = C 12 ε xx + C 11 ε yy − e 31 E z ,

(14) τ xy = 2 C 66 ε xy ,

(15) τ xxz = − f 14 E z ,

(16) τ yyz = − f 14 E z ,

(17) D z = e 31 ( ε xx + ε yy ) + ∈ 33 E z + f 14 ( η xxz + η yyz ) .

As strains in directions apart from thickness direction is negligible, they are not considered in the equation for simplification. The last term of Eq. (17) gives the electric field generated due to strain gradients developed in the nanoplates when no electric field is applied externally.

According to Gauss’ law of electrostatics, the electric displacement of the BNS-reinforced nanoplate without free electric charges is given by,

(18) ∂ D z ∂ z = 0 .

When the open circuit condition is considered for the nanoplate, the electric displacement on its surface is considered to be zero. Therefore, using Eq. (18), the internal electric field is derived as

(19) E z = e 31 ∈ 33 ∂ 2 w ∂ x 2 + ∂ 2 w ∂ y 2 z + f 14 ∈ 33 ∂ 2 w ∂ x 2 + ∂ 2 w ∂ y 2 ,

where the term e 31 ∈ 33 is concerned with the piezoelectricity and conveys electric field induced due to applied elastic strain and the term f 14 ∈ 33 is related with the flexoelectricity and exhibits electric field induced due to applied strain gradients.

When the surface effects are accounted, then the surface electric Gibbs free energy U s can be formulated as

(20) U s = − 1 2 ∈ γ κ s E γ s E κ s + 1 2 C α β γ κ s ε α β s ε γ κ s − e κ α β s ε α β s E κ s ,

where ∈ γ κ s is the dielectric constant linked with the surface, C α β γ κ s is the elastic constant for the surface, e κ α β s is the piezoelectric tensor of surface, ε α β s is the surface strain, and E κ s is the surface electric field. The linear surface constitutive relation incorporating surface effect can be expressed as

(21) σ α β s = ∂ U s ∂ ε α β = C α β γ κ s ε γ κ s − e κ α β s E κ s ,

(22) D γ s = − ∂ U s ∂ E γ s = ∈ γ κ s E κ s + e γ α β s ε α β s ,

wherein σ α β s is the Cauchy stress for surface, and D γ s represents the electric displacement related to the surface. Therefore, for the surface, the electric Gibbs free energy can be given as

(23) U s = 1 2 σ α β s ε α β s − 1 2 D γ s E γ s .

Based on the surface piezoelectricity model [17], for the surface layers, the surface constitutive are given as

(24) σ xx s = C 11 s ε xx + C 12 s ε yy − e 31 s E z ,

(25) σ yy s = C 12 s ε xx + C 11 s ε yy − e 31 s E z ,

(26) τ xy s = 2 C 66 s ε xy ,

where σ xx s , σ yy s , τ xy s , are the surface stresses, C 11 s , C 12 s , C 66 s are the surface elastic constants, and e 31 s is the surface piezoelectric constant.

The lateral displacement after incorporation of the flexoelectric and surface effects is derived by utilizing Hamilton’s principle [48]

(27) − δ ∫ t 1 t 2 ∫ V U d V + K + W d t = 0 ,

where U is the electric Gibbs free energy density including components of bulk and surface. Therefore, we can write U = U b + U s , furthermore, in the open circuit case, U can be given as

(28) U = 1 2 σ ij ε ij + 1 2 τ ijk η ijk .

When the vibration along the horizontal plane is neglected, the kinetic energy (K) can be written as

(29) K 1 2 ∫ V ρ ∂ w ∂ t 2 d V ,

where ρ indicates the mass density.

When the mechanical load is applied, the work done by it is defined as

(30) W = ∫ 0 a ∫ 0 b q 0 w d y d x .

Therefore, the governing equation can be expressed as

(31) ∂ 2 M xx ′ ∂ x 2 + ∂ 2 M xy ′ ∂ x ∂ y + ∂ 2 M yx ′ ∂ x ∂ y + ∂ 2 M yy ′ ∂ y 2 + ∂ 2 N xxz ′ ∂ x 2 + ∂ 2 N yyz ′ ∂ y 2 − ρ h ∂ 2 w ∂ t 2 + q 0 = 0 .

On the application of SS boundary condition on BNS reinforced nanoplate and considering the flexoelectricity and surface effects on general resultant moments and forces, the governing Eq. (31) can be reformulated and given by

(32) D 11 ∂ 4 w ∂ x 4 + ∂ 4 w ∂ y 4 + 2 ( D 12 + 2 D 66 ) ∂ 4 w ∂ x 2 ∂ y 2 + ρ h ∂ 2 w ∂ t 2 = q 0 ,

with

(33) D 11 = C 11 + e 31 2 ∈ 33 h 3 12 + C 11 s + e 31 s e 31 ∈ 33 h 2 2 + f 14 2 ∈ 33 h D 12 = C 12 + e 31 2 ∈ 33 h 3 12 + C 12 s + e 31 s e 31 ∈ 33 h 2 2 + f 14 2 ∈ 33 h D 66 = C 66 h 3 12 + C 66 s h 2 2 .

3 Electromechanical response of the BNS-reinforced nanoplate

The governing differential equation used in the analysis of the static bending analysis of the nanoplate is written in the modified form as

(34) D 11 ∂ 4 w ∂ x 4 + ∂ 4 w ∂ y 4 + 2 ( D 12 + 2 D 66 ) ∂ 4 w ∂ x 2 ∂ y 2 = q 0 .

As the nanoplate considered in this study is simply supported over all its edges, the solution for Eq. (34) can be obtained using Navier’s method. Utilizing the Navier’s method, the deflection of the plate in the transverse direction is expressed as

(35) w ( x , y ) = ∑ m = 1 ∞ ∑ n = 1 ∞ A mn sin α x sin β y ,

where A mn stands for the Fourier coefficient and depends on the positive integer-value and number of half waves m and n. Also, the values of α = m π / a and β = n π / b , should be used. The mechanical load applied on the nanoplate can be expressed in the form of Fourier series as

(36) q ( x , y ) = ∑ m = 1 ∞ ∑ n = 1 ∞ Q mn sin α x sin β y ,

where

A mn = Q mn d mn .

For hydrostatic loading,

(37a) Q mn = 8 q 0 cos ⁡ ( m π ) π 2 mn m , n = 1 , 3 , 5 , …

For point loading,

(37b) Q mn = 4 q 0 ab sin m π x 0 a sin n π y 0 b m , n = 1 , 2 , 3 …

For line loading,

(37c) Q mn = 8 q 0 π an sin m π x 0 a m = 1 , 3 , 5 , … n = 1 , 2 , 3 , …

(38) d mn = π b 4 D 11 m a 4 + D 12 n b 4 + 2 ( D 12 + 2 D 66 ) mn ab 2 .

Using Eqs. (32), (36)–(38), the analytical solution for lateral deflection w ( x , y ) for a nanoplate is given as,

(39) w ( x , y ) = ∑ m = 1,3,5 … ∞ ∑ n = 1,3,5 … ∞ Q mn sin α x sin β y mn D 11 m a 4 + D 12 n b 4 + 2 ( D 12 + 2 D 66 ) mn ab 2 .

4 PBRA using MCS

The aim of designing a system or product is to ensure that it effectively accomplishes its intended functions while maintaining satisfactory performance criteria, where performance criteria can be expressed as a permissible degree of damage or tolerance established through an acceptable likelihood of encountering failure. The methodology proceeds with the following steps:

Step 1: Create a set of design points covering entire design space

This step generates a set of input design vectors from the design space (also referred to as the experimental region) to evaluate the limit state at each set of values. This process, commonly known as Design of Experiments in the literature, can be classified into physical experiments or computer-generated experiments depending on the type of simulation model – stochastic and/or deterministic.

For deterministic models, the response values are determined through mathematical relationships based on the input variables. As these models are devoid of random errors, they primarily exhibit system error (e.g., approximations inherent in the model that may deviate from the true response). To minimize system error, space-filling designs, such as LHS, is employed to evenly or uniformly distribute points across the design space. LHS is particularly preferred for computer experiments due to its ability to provide dense and uniform coverage of the range of each variable within the design space. Using the LHS procedure, a specified number of sampling points (64 samples in the present work) for fiber volume fraction are generated. The fiber volume fraction is utilized to predict effective elastic and piezoelectric constants using the Mori-Tanaka model. For the sake of brevity the model is not presented here, for more details regarding model, the work of Kundalwal and Gupta [32] can be referred. Using these data points, curve fitting equations are obtained for effective elastic and piezoelectric constants, and these questions are used in the classical plate theory model presented in Section 3. Each simulation of the Mori-Tanaka model takes approximately 10 CPU minutes; thus, the use of response surface using LHS significantly reduced the computational burden compared to exhaustive Monte Carlo sampling.

Step 2. Evaluate the performance-based reliabilities by employing MCS at the design points

The performance-based reliabilities can be classified according to the properties of the performance indexes into four types, i.e., bigger the better, smaller the better, nominal the best, and zero point proportional [49]. This study has been undertaken based on the “nominal the best” performance index. This type of index is shown in Figure 2, which implies that the supplied performance in terms of the bending response of nanobeams shall be closer to the target value. Then, the reliability is defined as the probability that the bending response of nanoplate is within the given target values, i.e., R = P [ L 1 ≤ Bending Response ≥ L 2 ] .

Figure 2

Probability density of the bending response of nanobeams.

The estimated probability of failures P f ∼ of nanobeam’s bending response is calculated by solving the multi-dimensional integration over the failure region, where the performance function or limit state function denoted as G ( X , Y ) takes values less than or equal to zero [41]. Mathematically P f ∼ is expressed as

(40) P f ∼ = ∫ G ( X , Y ) ≤ 0 f μ X , μ Y ( X , Y ) d X d Y ,

where f μ X , μ Y is the joint probability density function (PDF) of all the input random variables and parameters. It is to be noted that for the multidimensional integration of Eq. (40), numerical methods become necessary due to the absence of a readily available explicit expression of joint PDF. Therefore, in this study, the MCS technique has been deployed by taking samples from the pdf of each design variable and problem parameter to acquire the PDF of the bending response. The unbiased estimation of P f ∼ can be obtained by

(41) P f ∼ = 1 N ∑ i = 1 N L ( X i , Y i ) = N f N ,

where N is the total number of samples for MCS, and N f is the total number of samples, where the value indicator L ( X i , Y i ) = 1 if G ( X , Y ) ≤ 0 .

5 Results and discussion

The BNS-reinforced nanocomposite material considered in this study consists of a polyimide matrix reinforced with piezoelectric BNNS. The mechanical and electrical properties are taken from our group’s previous study by Kundalwal and Gupta [32] and are presented in Table 1. These material properties are for 60% fiber volume fraction.

Table 1

Effective material properties of BNS-reinforced nanocomposite material

Material	Ref.	C 11 (GPa)	C 12 (GPa)	C 33 (GPa)	C 23 (GPa)	C 66 (GPa)	e 31 (C/m²)	e 33 (C/m²)	ϵ 33 (F/m)
Boron nitride reinforced nanocomposite	[32]	25.76	15.19	15.19	489.294	5.28	−0.0090	0.3192	1.38 × 10^–10

5.1 Effect of flexoelectricity and surface on the bending deflection of BNS-reinforced nanoplate

The variation in normalized bending stiffness ( D 11 / D 11 0 ) of BNS reinforced nanoplate with respect to its thickness h is displayed in Figure 3. D 11 and D 11 0 are the bending stiffnesses of the BNS-reinforced nanoplate with and without consideration of flexoelectricity, surface, and combined flexoelectricity and surface effect, respectively. It is clear from Eq. (33) that the bending stiffness depends only on the thickness of the nanoplate, and it is independent of its in-plane dimensions. It can be noted from Figure 3 that the stiffness of BNS-reinforced nanoplate with flexoelectric and surface consideration is respectively about four and six times greater than the bending stiffness of classical nanoplate, which does not consider flexoelectric and surface effect. When the flexoelectric and surface effect is considered together, there is a significant increase of ten times in the bending stiffness compared to the classical nanoplate case. As the thickness of the nanoplate is higher, the effect of both flexoelectricity and surface diminishes. For a smaller thickness of nanoplate, the effect of flexoelectricity and surface is significant, and hence, their effect cannot be ignored for a lesser value of thickness. This finding is consistent with the result described in the previous study [50]. Therefore, it is clear that the effect of surface and flexoelectricity on the normalized bending stiffness of BNS-reinforced nanoplate is size-dependent.

$Figure 3 Variation in normalized bending stiffness ( D 11 / D 11 0 {D}_{11}/{D}_{11}^{0} ) of BNS-reinforced nanoplate with respect to its thickness.$

Figure 3

Variation in normalized bending stiffness ( D 11 / D 11 0 ) of BNS-reinforced nanoplate with respect to its thickness.

To investigate the electromechanical response of the BNS-reinforced nanoplate, the results are obtained from the Kirchhoff plate theory modified to incorporate flexoelectric and surface effects. The length “a” and width “b” of the nanoplate are taken as 50 h and height h is taken as 3 nm. In this study, the aspect ratio of the plate is considered in the range of 5–80 as suggested by a previous study [51]. The load magnitude q ₀ is considered to be 0.05 MPa. The flexoelectric coefficient range for the polymers is mentioned between 10 − 8 and 10 − 9 C / m [52,53]. Hence, in this study, the flexoelectric coefficient is taken as 10^–9 C/m. The value of the surface elastic modulus and the coefficient of surface piezoelectricity is determined by assuming the surface layer thickness as 1 nm. Then, the surface properties are considered as the bulk properties of BNS-reinforced nanocomposite times the surface layer thickness [54,55]. The value of the half-wave numbers m and n are considered to be m = n = 1 in predicting all the results.

The effect of surface and flexoelectricity on the static bending performance of the 3 nm thick SS BNS reinforced nanoplate under point load, line load, and hydrostatic load is presented in Figures 4–6. All loads applied to the nanoplate are normal to the x-y plane. It can be seen from Figures 4–6 that the maximum deflection of the nanoplate occurs at the center of the nanoplate irrespective of the loading case. The deflection is observed to be maximum for the classical plate in all loading cases, where the effect of surface and flexoelectricity is not considered. When surface and flexoelectric effects are included in the nanoplate, there is a significant drop in maximum deflection, and it is independent of loading conditions. For example, in the point-loaded BNS reinforced nanoplate of 4 nm thickness, the reduction in maximum deflection is about 25% due to the flexoelectric effect, 75% due to the surface effect, and a significant 80% reduction can be observed when combined surface and flexoelectric effects are considered against maximum deflection of the classical plate. Size-dependent surface and flexoelectric effects are primarily responsible for this reduction in maximum deflection. The magnitude of the maximum deflection is highest in the point loading followed by line loading and lowest in the hydrostatic loading case.

Figure 4

Lateral deflection of SS BN-reinforced nanoplate with aspect ratio subjected to point load.

Figure 5

Lateral deflection of SS BNS-reinforced nanoplate with aspect ratio subjected to line load.

Figure 6

Lateral deflection of SS BNS-reinforced nanoplate with aspect ratio subjected to hydrostatic load.

Moreover, the maximum deflection is analyzed for the nanoplate thickness of 2, 4, 6, 8, and 10 nm in all three loading conditions to observe the effect of nanoplate thickness on the maximum deflection. The results are displayed in Table 2. In comparison to the classical model, for point loading with flexoelectric and surface effects, the maximum deflection is reduced by ∼80%, ∼63%, ∼53%, ∼45%, and ∼44% corresponding to 2, 4, 6, 8, and 10 nm, respectively. The maximum deflection amount reduces when the thickness of the nanoplate increases because the surface and flexoelectric effects are less prominent higher thickness. Similar observations are found for line and hydrostatic loading conditions.

Table 2

Maximum lateral deflection of SS BNRC nanoplate under different loading conditions

Thickness h (nm)	Case	Maximum deflection (nm)
Thickness h (nm)	Case	Uniformly distributed load	Hydrostatic load	Point load	In-line load
2	With flexo + surf	0.2004	0.1002	0.4944	0.3147
	With surf	0.2497	0.1248	0.6160	0.3922
	With flexo	0.5034	0.2517	1.2420	0.7907
	Classical	0.9986	0.4993	2.4640	1.5690
4	With flexo + surf	0.9503	0.4752	2.3450	1.4930
	With surf	1.0440	0.5219	2.5760	1.6400
	With flexo	2.0940	1.0470	5.1680	3.2900
	Classical	2.6100	1.3050	6.439	4.0990
6	With flexo + surf	1.42	0.7102	3.504	2.231
	With surf	1.498	0.749	3.696	2.353
	With flexo	2.701	1.35	6.664	4.242
	Classical	2.996	1.498	7.392	4.706
8	With flexo + surf	1.733	0.8663	4.275	2.721
	With surf	1.793	0.8967	4.425	2.817
	With flexo	2.957	1.487	7.295	4.644
	Classical	3.138	1.569	7.744	4.93
10	With flexo + surf	1.956	0.9778	4.485	3.072
	With surf	2.004	1.002	4.944	3.147
	With flexo	3.084	1.542	7.61	4.845
	Classical	3.206	1.603	7.91	5.036

We have also studied the effect of variation in the aspect ratio (x/a) on the maximum deflection of the nanoplate in all three loading conditions. For this purpose, the in-plane dimensions of the plate are kept constant as a = b = 100 nm. It can be noted from Figures 7–9 that for all loading conditions, the surface and flexoelectric effects are prominently seen for the lower nanoplate thickness, as both of these effects are size-dependent. As the thickness of the nanoplate increases, the effects of surface and flexoelectricity start diminishing and approach toward the results given for the classical plate case. It can be observed from these figures that the nanoplate subjected to point loading shows maximum defection (Figure 7), followed by line and hydrostatic loading conditions, respectively, as shown in Figures 8 and 9.

Figure 7

Maximum lateral deflection of SS BNS-reinforced nanoplate with aspect ratio h = a / x under point loading.

Figure 8

Maximum lateral deflection of SS BNS-reinforced nanoplate with aspect ratio h = a / x under line loading.

Figure 9

Maximum lateral deflection of SS BNS-reinforced nanoplate with aspect ratio h = a / x under hydrostatic loading.

The maximum deflection of the 3 nm fixed thickness nanoplate is also predicted and represented in Figures 10–12. We can note that the contribution of the flexoelectric effect in maximum deflection reduction is lesser compared to the surface effect. But still, the flexoelectric effect contribution cannot be ignored. The results presented in this study highlight the significance of the size-dependent effect and aspect ratio for an accurate model of a nanostructure.

Figure 10

Maximum lateral deflection of SS BNS-reinforced nanoplate with aspect ratio a = hx under point loading.

Figure 11

Maximum lateral deflection of SS BNS-reinforced nanoplate with aspect ratio a = hx under line loading.

Figure 12

Maximum lateral deflection of SS BNS-reinforced nanoplate with aspect ratio a = hx under hydrostatic loading.

5.2 Influence of uncertainty parameters on the bending response of nanoplate

This section discusses the uncertainty parameters and their effects on the bending response of nanoplate for various loading conditions such as point, line, and hydrostatic loading. Table 3 tabulates the distributional properties of geometrical parameters and variables. We also considered that the volume fraction of the nanoplate is driven by uncertainty, and hence, using LHS, the random values of volume fraction were obtained following the normal distribution. Utilizing these random values of volume fraction, the elastic and piezoelectric constants are evaluated using the Mori-Tanaka approach [32]. Later, the polynomial relation is obtained using the curve fitting data as shown in Table 4.

Table 3

Distributional properties of geometrical parameters and variables

Variables/parameters	Distribution	Mean (nm)	Standard deviation (nm)
H	Normal	3 × 10 − 9	0.1 × 10 − 9
a, b	Normal	150 × 10 − 9	0.2 × 10 − 9
Q	Normal	− 0.05 × 10 6	− 0.001 × 10 6
x	Normal	0.6	0.01
Volume fraction	Normal	0.6	0.01

Table 4

Curve fitting equations of material properties using LHS

Material constant	Distribution	Equations	Constant	SSE	R ²	Ad. R ²
C 11	Normal	C 11 = C 11 ′ x 2 + C 11 ″ x + C 11 ‴	C 11 ′ = 135 , C 11 ″ = − 107.1 , C 11 ‴ = 37.89	0.0025	1	1
C 12	Normal	C 12 = C 12 ′ x 2 + C 12 ″ x + C 12 ‴	C 12 ′ = 88.69 , C 12 ″ = − 70.16 , C 12 ‴ = 24.93	0.0011	1	1
C 66	Normal	C 66 = C 66 ′ x 2 + C 66 ″ x + C 66 ‴	C 66 ′ = 23.17 , C 66 ″ = − 18.49 , C 66 ‴ = 6.48	7.7 × 10⁻⁵	1	1
e 31	Normal	e 31 = e 31 ′ x 2 + e 31 ″ x + e 31 ‴	e 31 ′ = − 0.05139 , e 31 ″ = − 0.03982 , e 31 ‴ = − 0.01068	5.0 × 10⁻⁸	0.998	0.998
e 33	Normal	e 33 = e 33 ′ x 2 + e 33 ″ x + e 33 ‴	e 33 ′ = − 0.03603 , e 33 ″ = 0.6181 , e 33 ‴ = − 0.008751	5.3 × 10⁻⁸	1	1

SSE: Sum of squared errors; R ²: Coefficient of determination; Ad. R ²: Adjusted R ².

For this study, we used a sample size of 10,000. Since the original model is based on an exact solution, the computational cost per sample is relatively low. The computational cost of solving the bending response of a flexoelectric plate using the Kirchhoff plate theory was approximately 10–15 CPU seconds per simulation. Using MATLAB solver R2021a on a multi-core processor, the total computational cost for generating 10,000 samples was approximately 10 CPU minutes, demonstrating the efficiency of our approach. This highlights the computational efficiency of the Monte Carlo method in our study, even for a large number of simulations.

The performance of a design is usually represented with many measurable engineering specifications. These specifications are customarily marked either with a value holding tolerance or with a range having lower and upper limits. Figure 13(a) illustrates the PDF of point load with limit state function, which, in our case, is the deflection (w). The figure depicts that the performance of the piezoelectric BNRC nanocomposite is greatly affected by the uncertainty in the geometric parameters, fiber volume fraction, and applied load while designing the piezoelectric nanocomposite plate for the application of distributed sensors and actuators. The performance-related reliability of the system can be considered by choosing a zero-point proportional performance index. This type involves the output and the input being closer to a constant proportional as the better. The ideal quality may be a line across the origin. Similarly, the PDF and reliability are plotted for line and hydrostatic loading as shown in Figures 14 and 15, respectively.

Figure 13

(a) PDF of limit state function and (b) variation in reliability with limit state function for point loading.

Figure 14

(a) PDF of limit state function and (b) variation in reliability with limit state function for line loading.

Figure 15

(a) PDF of limit state function and (b) variation in reliability with limit state function for hydrostatic loading.

Figure 13(b) shows the variation in the reliability with the limit state function for point loading. It can be seen that 90% reliability is obtained when deflection is ( 1.52 ± 0.19 ) × 10 − 9 , which is desirable reliability. As the deflection of the plate deviates from ( 1.52 ± 0.19 ) × 10 − 9 , the reliability index starts decreasing and approaches zero. Similarly, corresponding to line loading, 90% reliability is obtained at ( 9.7 ± 0.12 ) × 10 − 10 and for hydrostatic loading, 90% reliability is obtained at ( 3.1 ± 0.36 ) × 10 − 10 (Figures 14(b) and 15(b), respectively).

6 Conclusion

The static bending behavior of the SS BNS-reinforced nanoplate is investigated under hydrostatic load, point load, and line load case. The Navier solution method is applied to develop the closed-form solution for the governing differential equation. Surface and flexoelectric effects are incorporated in the Kirchhoff plate model, and their contribution to static bending behavior is studied. It is observed that maximum deflection occurs at the center of the nanoplate in all loading cases. The surface effect, flexoelectric effect, and integrated surface and flexoelectric effect reduce the maximum bending of the nanoplate. It is also noted that the flexoelectric and surface effects are more prominent for the smaller thickness of the nanoplate, and their effect diminishes as the plate thickness increases. Therefore, surface and flexoelectric effects are size-dependent and must be included in the model to predict the static bending response. The performance-based reliability is obtained using MCS and the results present 90% reliability for various loading conditions. The insights from this work can have practical applications in the design and optimization of advanced nanodevices. For instance, the enhanced sensitivity to flexoelectric and surface effects can be utilized to develop high-performance sensors that respond to minute mechanical stimuli. Similarly, the ability to fine-tune the electromechanical properties through material and structural parameters makes these nanoplates promising candidates for actuators, where precise control of deformation is required. Furthermore, the inclusion of flexoelectric effect can enhance the energy conversion efficiency in energy harvesting devices, enabling the development of compact and efficient systems for powering nanoscale electronics.

Acknowledgments

This work was generously supported by the Science Engineering Research Board (SERB), Department of Science and Technology, Government of India. The authors acknowledge the financial support of the SERB (grant number CRG/2022/000786, PDF/2021/000225).

Funding information: The authors would like to thank the Science Engineering Research Board (SERB), Department of Science and Technology, Government of India for providing financial support.
Author contributions: M. Gupta: data curation, formal analysis, investigation, methodology, software, validation, visualization, and writing – original draft. P.V. Bansod: formal analysis, investigation, software, validation, visualization, and writing – original draft. H. Gargama: formal analysis, investigation, software, validation, visualization, and methodology. S.I. Kundalwal: conceptualization, funding acquisition, investigation, methodology, project administration, resources, supervision, and writing – review and editing. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: Authors state no conflict of interest.
Data availability statement: All data generated or analyzed during this study are included in this published article.

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Received: 2024-09-30

Revised: 2025-01-09

Accepted: 2025-02-10

Published Online: 2025-05-16

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

Articles in the same Issue

https://doi.org/10.1515/jmbm-2025-0045

Keywords for this article

boron nitride-reinforced composite; Navier solution; flexoelectricity; surface effect; nanoplate; performance-based reliability; Monte Carlo simulation

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