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We investigated the relationship between porosity and flexoelectricity for aluminum (Al)/polyvinylidene fluoride (PVDF) composites. Neat PVDF, composites of micron aluminum (μAl)/PVDF, and composites of nano aluminum (nAl)/PVDF were 3D printed, and the flexoelectric response was measured using a cantilever beam test setup. Voids (up to 72.4 mm3) were incorporated into the samples by decreasing the infill percent of the 3D printed material. We found that increasing the porosity via millimeter scale voids incorporated into the infill pattern decreased the average effective flexoelectric coefficient relative to the near full-density (100% infill) control samples. This contrasts with other studies that have shown increasing micron scale porosity increases the flexoelectric coefficient. In addition, we measured higher flexoelectric responses for nAl/PVDF than μAl/PVDF as well as for samples printed by the Hyrel 3D SR printer as opposed to the Ender 3 V2 printer. These results indicate that charge generation due to flexoelectricity can be altered by changing parameters such as porosity, particle size of inclusions, or manufacturing method. Smaller voids and fine particles can induce larger strain gradients than larger inhomogeneities, leading to increased flexoelectric coefficients. A competing effect is that more porosity leads to less materials, which can decrease the flexoelectric coefficient. [ABSTRACT FROM AUTHOR]

We report a theoretical modeling of the flexoelectric response of dielectric–ferroelectric (DE–FE) multilayers based on phase field simulations in the framework of the Landau–Ginzburg–Devonshire (LGD) theory. The correlation between negative capacitance and flexoelectric response is revealed, and the single-domain and multi-domain models are compared. It shows that the dielectric layers drive the ferroelectric layer into a negative capacitance regime, and the flexoelectric response of the multilayer is maximal when the negative capacitance of the ferroelectric layer has a minimal absolute value. Moreover, the flexoelectric response peak will be shifted to a lower temperature by increasing the thickness of dielectric layer, indicating a possibility of achieving a stronger flexoelectric response at room temperature compared with that of pure ferroelectric. However, while the single-domain model shows that the flexoelectric response peak is simply shifted to a lower temperature with near constant peak value and width, the multi-domain model reveals a significant suppressing of the flexoelectric peak by the dielectric layer. This is attributed to the formation of the vortex domain state, which eases the depolarization effect and leads to large absolute value of negative capacitance of the ferroelectric layer. Our work provides new insights into flexoelectricity in ferroelectric heterostructures. [ABSTRACT FROM AUTHOR]

Poly(vinylidene fluoride) (PVDF) has been widely studied because of its various crystal phase transformations as well as its special dielectric, piezoelectric, and other electrical properties. In this study, a temperature gradient (T-gradient) was constructed for studying the temperature dependence of the crystallization behavior of PVDF and the heterogeneous nucleation effect of multiwall carbon nanotubes (MWCNTs). The nanoscale structure of PVDF was investigated through three different dimensions using FTIR for the surface, WAXD/SAXS for the bulk and AFM for the cross-section. The results showed that low temperature and a heterogeneous nucleating agent were both conducive to the formation of the γ phase of PVDF. The distance of MWCNT nucleation effect is about 15 μm, and it is more significant in 0–5 μm. The content of the γ phase reached 77% when Tc was lower than 61 °C. The higher content and lower crystallite size of the γ phase lead to a higher dielectric constant, which can lay a foundation for its future application in the field of dielectrics. [ABSTRACT FROM AUTHOR]

The modified strain gradient theory (MSGT) with flexoelectric and micro-inertia effects is developed to study the free vibration of tube made of elastic centrosymmetric materials. Within this theory, the internal energy density is the function of the strain tensor, deviatoric part of stretch gradient and rotation gradient tensors, dilatation gradient, polarization vector, and electric quadrupoles. The linear coupled governing equations and boundary conditions are derived from the Hamilton variation principle. The torsional vibration problem is analytically solved for non-piezoelectric nanotube with fixed edges. The effect of the nanotube geometry, flexoelectric coefficient, micro-inertia, and micro-stiffness parameters on the natural frequency is studied. [ABSTRACT FROM AUTHOR]

[Display omitted] Deep-level traps at the buried interface of perovskite and energy mismatch problems between the perovskite layer and heterogeneous interfaces restrict the development of ideal homogenized films and efficient perovskite solar cells (PSCs) using the one-step spin-coating method. Here, we strategically employed sparingly soluble germanium iodide as a homogenized bulk in-situ reconstruction inducing material preferentially aggregated at the perovskite buried interface with gradient doping, markedly reducing deep-level traps and withstanding local lattice strain, while minimizing non-radiative recombination losses and enhancing the charge carrier lifetime over 9 µs. Furthermore, this gradient doping assisted in modifying the band diagram at the buried interface into a desirable flattened alignment, substantially mitigating the energy loss of charge carriers within perovskite films and improving the carrier extraction equilibrium. As a result, the optimized device achieved a champion power conversion efficiency of 25.24% with a fill factor of up to 84.65%, and the unencapsulated device also demonstrated excellent light stability and humidity stability. This work provides a straightforward and reliable homogenization strategy of perovskite components for obtaining efficient and stable PSCs. [ABSTRACT FROM AUTHOR]

Flexoelectricity is the intrinsic length‐scale dependent higher‐order electromechanical response of all centro‐ and non‐centro‐symmetric dielectrics, including piezoelectrics. Direct flexoelectricity is defined as the appearance of an electric field due to induced strain gradients. The numerical modeling of flexoelectricity is largely carried out using mixed FE, which has its historical foundations in strain gradient theories. However, existing finite elements are either limited to 2D or have inherited numerical instabilities due to the known saddle‐point structuring. The current work presents a numerically robust three‐dimensional mixed FE for higher‐order electromechanical applications without the use of stabilization or penalty parameters. After its verification, the new finite element is applied to the new problem of truncated semicone torsion, taking into account flexoelectricity in piezoelectric solids, and the original findings are reported. Current research reveals the complex interaction between first‐order (piezoelectricity) and higher‐order (flexoelectricity) electromechanical coupling. [ABSTRACT FROM AUTHOR]

Flexoelectricity, a universal electromechanical coupling phenomenon, has triggered new feasibilities of advancements in functional materials, especially for nanoscale materials. The strong flexoelectric response is initially discovered in ceramic materials with high permittivity, and then the past decades have witnessed the expansion of flexoelectricity to a broader range of material systems including semiconductors, polymers, and soft elastomers, which in turn raise emerging applications of flexoelectricity. Moreover, flexoelectricity is demonstrated to be significantly enhanced in thin films and nanostructures where ultra‐high strain gradients are easier to achieve, rendering flexoelectricity attractive for modifying the functional properties of advanced materials and devices at the nanoscale. To provide a comprehensive drawing of the above aspects, this review highlights the recent progress of flexoelectricity in diverse materials, covering the characterization of flexoelectricity, the fundamental mechanisms of the enhancement flexoelectric response as well as the multi‐functional applications. Finally, some open questions and perspectives are presented, underlining the fascinating future of this field. [ABSTRACT FROM AUTHOR]

The piezoelectric and flexoelectric techniques are of great interest among the scientific fraternity, especially for the design of vibration-based micro/nano energy harvesters. The development of such energy harvesters in the micro/nanoscale has a huge potential, mostly in wireless networks and IoT-based applications. The present work thus formulates novel nanostructure-based analytical models of energy harvesters with asymmetric proof mass considering nonlocal flexoelectricity, geometrical nonlinearity, mid-plane stretching, harmonic base, and axially restraint forces. Using non-classical Euler-Bernoulli nanobeam and Hamilton's principle, governing differential equation of motion has been derived. Analytical solutions have been obtained to compute the eigenfrequencies and subsequent modes of vibration concerning various systems' physical and geometric variables. Nonlinear steady-state voltage and power characteristics thereafter have been investigated and studied their local stability and bifurcation for primary and sub-harmonic resonant conditions. The influences of performance factors associated with asymmetrically proof mass, nonlocal elasticity, and time-dependent base and restraint forces on the nonlinear voltage and power output using frequency characteristics criterion. These results have been validated with the findings using COMSOL-2D, and interestingly, mean voltage and power responses from linear models are found to be highly consistent with findings in COMSOL 2D. The present outcome shows how the asymmetric proof mass and nanoscale effect create disparities in harvesting energy performance and, simultaneously, how important their role is to avoid any undesirable structural instability. The present studies together can strengthen the significance of size-dependency and asymmetric proof mass while designing a nanoscale generator in the presence of harmonic excitation. [ABSTRACT FROM AUTHOR]

Biopolymers, the potential flexoelectric materials, are environment-friendly, degradable, lightweight, cost-effective, and possess remarkable processing properties catering to the requirements of advanced devices. However, the flexoelectric coefficient of biopolymers is normally much weaker than that of ceramic materials, limiting their potential applications for designing high-performance green electromechanical coupling devices. To improve the flexoelectric response in biopolymers, we composited barium titanate (BTO) with 2,2,6,6-tetramethylpiperidine-1-oxyl -oxidized cellulose nanofibrils (TOCNF) to enhance the flexoelectric response of TOCNF. Owing to the high permittivity and flexoelectric effect of BTO, the relative dielectric constant and flexoelectric coefficient of 33.3 wt% BTO-TOCNF films reached 30.94 @ 1 kHz and 50.05 ± 1.88 nC/m @ 1 Hz, which were almost 172 times and 27 times higher than those of TOCNF, respectively. The composite thin film contains high dielectric constant and flexoelectric coefficient, as well as excellent flexibility. Our study provided a straightforward and efficient method for improving the flexoelectric effect of biopolymers, and demonstrated its great potential applications in flexoelectric-based devices. [ABSTRACT FROM AUTHOR]

Flexoelectricity is a two-way coupling effect between the strain gradient and electric field that exists in all dielectrics, regardless of point group symmetry. However, the high-order derivatives of displacements involved in the strain gradient pose challenges in solving electromechanical coupling problems incorporating the flexoelectric effect. In this study, we formulate a phase-field model for ferroelectric materials considering the flexoelectric effect. A four-node quadrilateral element with 20 degrees of freedom is constructed without introducing high-order shape functions. The microstructure evolution of domains is described by an independent order parameter, namely the spontaneous polarization governed by the time-dependent Ginzburg–Landau theory. The model is developed based on a thermodynamic framework, in which a set of microforces is introduced to construct the constitutive relation and evolution equation. For the flexoelectric part of electric enthalpy, the strain gradient is determined by interpolating the mechanical strain at the node via the values of Gaussian integration points in the isoparametric space. The model is shown to be capable of reproducing the classic analytical solution of dielectric materials incorporating the flexoelectric contribution. The model is verified by duplicating some typical phenomena in flexoelectricity in cylindrical tubes and truncated pyramids. A comparison is made between the polarization distribution in dielectrics and ferroelectrics. The model can reproduce the solution to the boundary value problem of the cylindrical flexoelectric tube, and demonstrate domain twisting at domain walls in ferroelectrics considering the flexoelectric effect. [ABSTRACT FROM AUTHOR]

This paper presents isogeometric analysis (IGA)-based topology optimization for electrical performance of three-dimensional (3D) flexoelectric structures. IGA is employed to provide C 1 continuity in shape function, which is required in treating high-order electromechanical coupling equations. To improve the computational efficiency in treating 3D problems, the redundant degrees of freedom removal technique is introduced. Regularization treatments are also implemented to avoid the numerical singularity induced by flexoelectricity. Both virtual loads and strain energy constraints are taken into consideration to prevent unexpected structural disconnection. Numerical examples and experiments on optimized structures demonstrate that the flexoelectric performance can be effectively improved using the proposed approach. [ABSTRACT FROM AUTHOR]

Flexoelectricity is present in nonuniformly deformed dielectric materials and has size-dependent properties, making it useful for microelectromechanical systems. Flexoelectricity is small compared to piezoelectricity; therefore, producing a large-scale flexoelectric effect is of great interest. In this paper, we explore a way to enhance the flexoelectric effect by utilizing the snap-through instability and a stiffness gradient present along the length of a curved dielectric plate. To analyze the effect of stiffness profiles on the plate, we employ numerical parameter continuation. Our analysis reveals a nonlinear relationship between the effective electromechanical coupling coefficient and the gradient of Young's modulus. Moreover, we demonstrate that the quadratic profile is more advantageous than the linear profile. For a dielectric plate with a quadratic profile and a modulus gradient of − 0.9, the effective coefficient can reach as high as 15.74 pC/N, which is over three times the conventional coupling coefficient of piezoelectric material. This paper contributes to our understanding of the amplification of flexoelectric effects by harnessing snapping surfaces and stiffness gradient design. [ABSTRACT FROM AUTHOR]

The flexoelectric effect describes the interplay between material polarization and strain gradient. While much research has focused on control strategies for flexoelectricity, the impact of ferroelectric domain structure on the flexoelectric effect remains elusive. Through phase field simulations, we conduct a comparative study on the flexoelectric response of two types of BaTiO3–SrTiO3 binary films: BaTiO3–SrTiO3 multilayers and (Ba,Sr)TiO3 solid solutions. Our findings reveal that the flexoelectric response of these two systems, at the same Ba/Sr ratio, differs not only in magnitude but also in other aspects like temperature dependence, linearity, and hysteresis, due to their different ferroelectric domain structures and evolution dynamics. Our study deepens our current understanding of flexoelectricity in ferroelectrics and suggests "domain engineering" as a potential way to tailor this effect. [ABSTRACT FROM AUTHOR]

Based on the three-dimensional (3D) basic equations of piezoelectric semiconductors (PSs), we establish a two-dimensional (2D) deformation-polarization-carrier coupling bending model for PS structures, taking flexoelectricity into consideration. The analytical solutions to classical flexure of a clamped circular PS thin plate are derived. With the derived analytical model, we numerically investigate the distributions of electromechanical fields and the concentration of electrons in the circular PS thin plate under an upward concentrated force. The effect of flexoelectricity on the multi-field coupling responses of the circular PS plate is studied. The obtained results provide theoretical guidance for the design of novel PS devices. [ABSTRACT FROM AUTHOR]

Piezoelectric semiconductors (PSCs) find extensive applications in modern smart electronic devices because of their dual properties of being piezoelectric and semiconductive. With the increasing demand for miniaturization of these devices, the performance of their components needs to be carefully designed and optimized, especially when reduced to nanosize. It has been shown that surface elastic properties play a substantial role in the mechanical performance of nanoscale materials and structures. Building on this understanding, the surface elastic effects, encompassing surface residual stress, surface membrane stiffness, and surface bending stiffness, are comprehensively taken into account to explore the electromechanical responses of a PSC nanobeam. Additionally, the flexoelectric effect on their responses is also systematically studied. The results of this work reveal that surface elastic properties predominantly influence mechanical performance, while the flexoelectric effect plays a more dominant role in electric-related quantities at the nanoscale. Notably, the significance of surface bending rigidity, which was often underestimated in the earlier literature, is demonstrated. Furthermore, owing to the flexoelectric effect, the linear distribution of electric potential and charge carriers along the length transforms into a nonlinear pattern. The distributions of electric potential and charge carriers across the cross section are also evidently impacted. Moreover, the size-dependent responses are evaluated. Our findings may provide valuable insights for optimizing electronic devices based on nanoscale PSCs. [ABSTRACT FROM AUTHOR]

The functionally graded (FG) flexoelectric material is a potential material to determine the structural morphing of aircrafts. This work proposes the penalty 20-node element based on the consistent couple stress theory for analyzing the FG flexoelectric plate and shell structures with complex geometric shapes and loading conditions. Several numerical examples are examined and prove that the new element can predict the size-dependent behaviors of FG flexoelectric plate and shell structures effectively, showing good convergence and robustness. Moreover, the numerical results reveal that FG flexoelectric material exhibits better bending performance and higher flexoelectric effect compared to homogeneous materials. Moreover, the increase in the material length scale parameter leads to a gradual increase in the natural frequencies of the out-of-plane modes of FG flexoelectric plate/shell, while the natural frequencies of the in-plane modes change minimally, resulting in the occurrence of mode-switching phenomena. [ABSTRACT FROM AUTHOR]

The unique morphology of 2D van der Waals materials enables them to withstand large deformations and significant nonuniform strain, potentially inducing a strong flexoelectric effect. Despite the size‐dependent flexoelectric effect showing potential for modulating the optoelectronic performance of 2D van der Waals materials, it is far from being fully exploited owing to various challenges. Herein, the use of nanowires with different diameters as the bending media to fabricate 2D α‐In2Se3/β‐InSe heterojunctions with large curvatures of 0.1–1 µm−1 is proposed. The significant band alignment modulation in α‐In2Se3 resulting from the bending‐induced flexoelectric effect is verified through Kelvin probe force microscopy. The strain‐induced piezoelectric effect can be negated because of the weak vdW forces at the interface. The flexoelectric polarization in β‐InSe is screened via the accumulated electrons in the unilateral depleted heterojunction. Compared to the flat heterojunction, the curved heterojunction with an average curvature of 0.9 µm−1 shows 2.48‐fold and 7.62‐fold increases in open‐circuit voltage and zero‐biased responsivity, respectively. This study demonstrates the first successful modulation of photodetection in 2D heterojunctions by exploiting the flexoelectric effect, providing a new perspective for high‐performance 2D vdW optoelectronic devices. [ABSTRACT FROM AUTHOR]

Flexoelectricity is a universal phenomenon in all dielectrics, which couples electrical polarizations with strain gradients. However, limited by the size effects, the intensity of the flexoelectric effect becomes appreciable at the micro- and nano-scale, which suppresses its applications at the macroscale. To amend this inconvenience in applications, we innovatively propose replacing the traditional pure dielectric film with flexoelectret film containing quasi-permanent electrical charges in crumpling deformation. The crumpled flexoelectret film can give a considerable flexoelectric-like response at the macroscale. Specifically, we establish a theoretical model for the crumpled flexoelectret film, derive the analytical solution of the electromechanical coupling under the bi-directional bending conditions, and analyze the flexoelectric-like response under different thicknesses, charge densities, and scales. The results show that when the flexoelectret film's thickness H = 1 mm and the charge density q = − 0.2 mC ⋅ m − 2 , the intensity of its flexoelectric-like response is nearly 100 times of that of pure dielectric film's intrinsic flexoelectricity. [ABSTRACT FROM AUTHOR]

• Flexoelectricity and surface effect on piezoelectric nanolaminates were reported. • Analytical solution of bending deformation and vibration were obtained. • Effects of residual surface stress and the flexoelectric coefficient were examined. • Effects of aspect ratio and plate thickness on bending behavior were examined. In this paper, a model of double-layer rectangular piezoelectric nanolaminates with different upper and lower surface properties is established. The bending and vibration behavior of the piezoelectric nanolaminates are evaluated and analyzed, taking into consideration the flexoelectric and surface effects. Utilizing the surface piezoelectric model, flexoelectric theory, and Kirchhoff plate theory, this study provides analytical solutions for the bending deflection and natural frequency of rectangular piezoelectric nanolamination materials under various boundary conditions. The results indicate that the effects of flexoelectric and surface influences on piezoelectric nanolaminates are associated with residual surface stress and flexoelectric coefficients. These effects can enhance the bending stiffness and decrease the natural frequency of the piezoelectric nanolaminates. Furthermore, altering the residual stress on the upper and lower surfaces of the laminates separately will result in different trends in bending deflection, while causing a consistent trend in natural frequency change. When the electric potential direction of piezoelectric nanolaminates is different, the bending deflection of piezoelectric nanolaminates will change accordingly. The natural frequency of piezoelectric nanolaminates is determined by the absolute value of the flexoelectric coefficients, and is independent of their positive or negative values. Both the surface effect and the flexoelectric effect are influenced by size-dependent behaviors. When the thickness of the laminates is sufficiently large, both the surface effect and flexoelectric effect can be disregarded. This paper's research methods and results provide theoretical models and analytical methods for the microstructure design, multi-physical field characterization, and bending deformation behavior of intelligent components containing rectangular piezoelectric nanolaminates. [ABSTRACT FROM AUTHOR]

In spite of the flexoelectric effect being a universal phenomenon in the ferroelectric perovskites, the current understanding of flexoelectric aging in ferroelectrics is, actually, rather incomplete. In this paper, we have fabricated a series of Mn-doped BaTiO3 perovskite ceramics (BaTi1–xMnxO3, x = 0.1% and 1%, BTMO) to systematically investigate the corresponding flexoelectric aging behavior by controlling the concentration of Mn. We found that the variation of Mn dopant significantly effects the Curie temperature, dielectric constant, flexoelectric aging, and flexoelectric coefficient of the BTMO ceramics. Especially for the BTMO (0.1%) ceramics, obvious ferroelectric aging and flexoelectric aging phenomenon are observed at room temperature. The main reason for aging of BTMO ceramics is that the doping of Mn introduces oxygen vacancies, which tend to be stable under the action of strain gradient and electric field. Therefore, the results presented in this paper verify that the flexoelectric aging in Mn-doped BTO ceramics is closely related to ferroelectric fatigue. [ABSTRACT FROM AUTHOR]

AbstractThis article presents a size dependent mathematical model and an analytical solution methodology to accurately simulate and analyze the size dependent buckling behavior of piezoelectrically layered sandwich nanobeams with perforated core embedded in an elastic medium with flexoelectricity. The electric enthalpy energy function is expressed in terms of the electric and the flexoelectric effects. Regular squared cutouts perforation pattern is adopted for the perforated core. Closed forms for the equivalent geometrical parameters of perforated core are developed. The shear deformation effect is incroporated using the Timoshenko beam theory (TBT). To capture the nonlocality and the microstructure length scale effects, the nonlocal strain gradient elasticity theory is modified and adopted to include the electromechanical nonclassical effects. The Hamiltonian principle is utilized to derive the equilibrium equations. An analytical solution methodology is developed to derive closed forms for critical buckling loads. The developed solution procedure is implemented into a MATLAB software code. The accuracy of the developed procedure is verified by comparing obtained results with corresponding cases reported in the literature. Influences of different design variables on the electromechanical buckling behavior are explored through intensive parametric studies. Results obtained showed the significant effects of the flexoelectricity and the piezoelectric parameters on the size dependent buckling behavior of piezoelectric sandwich nanobeams with perforated core. Size dependent electromechanical as well as mechanical buckling behaviors could be controlled by adjusting these parameters. The developed procedure and the obtained numerical results are helpful in many technological and industrial applications as MEMS and NEMS. [ABSTRACT FROM AUTHOR]

In the present review, the merits and demerits of machine learning (ML) in materials science are discussed, compared with first principles calculations (PDE (partial differential equations) model) and physical or phenomenological ODE (ordinary differential equations) model calculations. ML is basically a fitting procedure of pre-existing (experimental) data as a function of various factors called descriptors. If excellent descriptors can be selected and the training data contain negligible error, the predictive power of a ML model is relatively high. However, it is currently very difficult for a ML model to predict experimental results beyond the parameter space of the training experimental data. For example, it is pointed out that all-dislocation-ceramics, which could be a new type of solid electrolyte filled with appropriate dislocations for high ionic conductivity without dendrite formation, could not be predicted by ML. The merits and demerits of first principles calculations and physical or phenomenological ODE model calculations are also discussed with some examples of the flexoelectric effect, dielectric constant, and ionic conductivity in solid electrolytes. [ABSTRACT FROM AUTHOR]

This study explores the incorporation of flexoelectricity and size-dependence effects in electricity harvesting from a novel nanostructure. This nanostructure, featuring a cylindrical-shaped rigid tip mass, is stimulated at the base with axial pulsating excitation. Designed with a nanobeam material, the energy harvester accounts for structural nonlinearity, flexoelectricity, and the presence of an off-center rigid mass, operating under multiple excitations. The eigenanalysis is employed to delve into the dynamic characteristics of the system, providing engineers with insights to identify safe operating regions and operational constraints. Additionally, perturbation techniques are utilized to estimate steady-state voltage and power characteristics, with a focus on maximizing voltage and power generation. The study underscores the impact of size dependence on nanoscale design, flexoelectricity, asymmetric tip mass, and various excitation parameters within a constrained operating range. Analysis reveals how saddle-node and pitchfork bifurcations can disrupt energy harvesting performance and suggests strategies for mitigating these issues by adjusting operational parameters. Moreover, the study emphasizes the significant role of the off-center rigid tip mass in energy estimation compared to conventional microstructure-based energy harvesters. Analytical findings are rigorously validated against numerical results under different resonant scenarios, showcasing the accuracy of the analytical approach in capturing nonlinear sources and optimizing energy harvesting from nanostructures. [ABSTRACT FROM AUTHOR]

AgNbO3 is a typical antiferroelectric ceramic with weak ferroelectricity at room temperature. Due to this feature, AgNbO3 has received a lot of attention in electromechanical applications, but most of these studies have focused on the doping of other elements in AgNbO3. In this study, curved AgNbO3 ceramics were fabricated using a gravity-driven annealing process, and the field-induced strains were investigated. The curved ceramic exhibited a significantly higher field-induced nominal strain compared to a flat ceramic. Experimental results revealed that the curved AgNbO3 sample exhibited a maximum nominal strain of 1.22%, significantly surpassing the theoretical value of 0.13%. Analysis of various characterizations confirmed the presence of flexoelectricity, and flexoelectricity played an important role in the ultrahigh electromechanical response of curved AgNbO3 ceramics. Additionally, we demonstrated that curved AgNbO3 ceramics can enhance the flexoelectric coefficient. These findings suggest that curved antiferroelectric ceramics can enhance their displacement output, meeting the requirements for applications in large-strain actuators. [ABSTRACT FROM AUTHOR]

Direct flexoelectricity is a size-dependent phenomenon, very prominent at smaller scales, that connects the strain gradients and the electric field. The very existence of strain gradients enhances noncentrosymmetry and heightens the interaction between piezoelectricity and flexoelectricity, demanding fully coupled higher-order electromechanical formulations. The numerical instability of the existing finite elements used to model flexoelectricity alone is revealed due to their reliance on the stabilization parameter. Thus, two new finite elements Qu2s2p2l0 (QL0-4) and Qu2s2p2l1 (QL1-16) are proposed for mixed FEM that are numerically robust without any need of such stabilization parameters. Additionally, the existing finite element Qu2s1p2l0 [Q47 in (Deng et al. in J Appl Mech 84:081004, 2017)], is implemented from scratch to replicate known results and benchmark the performance of newly proposed finite elements. To verify the robustness of these elements, various benchmark problems for flexoelectricity in dielectric solids, such as a thick cylinder and truncated pyramid are simulated. The great agreement of the numerical results with the existing ones reflects the foundational nature of the proposed elements. Furthermore, the proposed mixed finite elements were used to successfully analyze cantilever beam and truncated pyramid problems where piezoelectric effects were taken into account for the first time. Current results are intrumental in simulating flexoelectricity and piezoelectricity together to highlight their interactions using newly proposed numerically robust finite elements. [ABSTRACT FROM AUTHOR]

A nonlinear phenomenological model of flexoelectricity in thermoelastic solids is presented within the frame of continuum mechanics and extended thermodynamics, incorporating the quasi-electrostatic approximation where the time derivative of the electric displacement vector can be neglected in Maxwell–Ampère's law. An expression for the energy flux is proposed, including many internal variables. An advantage of the proposed model is that it presents a unified approach to study several thermo-electromechanical couplings, including electrostriction, piezoelectricity, dependence of entropy and spontaneous polarization on flexoelectricity, among others. The model may be of interest in revealing the effects of such couplings on the properties of polarizable media, in particular ferroelectrics, when simple boundary conditions prevail. For a fully dynamic description of flexoelectricity, Hamilton's principle and the variational principle for external forces should be used instead. [ABSTRACT FROM AUTHOR]

In this work, a novel multiscale topology optimization method has been proposed for the design of electromechanical energy-harvesting systems converting mechanical vibrations into electric currents made of non-piezoelectric materials. At the microscopic scale, the material is assumed to be periodic, porous, and flexoelectric, although not piezoelectric. A first step of topology optimization is performed, in order to maximize the effective (homogenized) flexoelectric properties of the material, where a flexoelectric homogenization model is first formulated. As a result, the effective material, although made of a non-piezoelectric material, has apparent piezoelectric properties. In a second step, these properties are used to model the behavior of a dynamic electromechanical energy-harvesting system structure. A second topology optimization step, this time performed at the structural scale, aims to maximize the system electromechanical coupling factor (ECF) for a given forced vibration frequency, including the micro-inertial effect. At both scales, an isogeometric analysis method is employed to solve the strain-gradient problems numerically. We show that the optimized structure obtained offers significant gains in terms of ECF (by a factor of between 2 and 20) compared with non-optimized structures of the same volume, over a wide range of excitation frequencies. The procedure could open up new possibilities in the design of energy recovery systems without the use of piezoelectric materials. [ABSTRACT FROM AUTHOR]

This paper studies electro-thermal buckling in a flexoelectric microbeam. This first-of-a-kind study focuses on two case-specific types of buckling in microbeams, which are essential to microstructures. Because of the electric fields and high temperatures in their environments, microbeams are susceptible to electro-thermal buckling more often than mechanical buckling. In addition, microstructures can exhibit flexoelectricity. As such a thorough understanding of electro-thermal bucking is extremely important. In this paper, we begin with extracting the equations governing flexoelectric microbeams from size-dependent flexoelectric theories. We then derive closed-form solutions for said equations. We explore various boundary conditions and, with respect to the load, analyze the various pre-buckling and buckling modes. Our study shows that buckling does not occur in some boundary conditions, and electrical curvature is very important in isotropic and homogeneous flexoelectric microbeams. In the end, the paper presents electro-thermal buckling results for specific boundary conditions as well the probability of buckling. [ABSTRACT FROM AUTHOR]

AbstractFlexoelectricity indicates a unique characteristic of insulators and dielectric materials that results in electrical polarization induction in response to the non-uniform strain gradient. The impact of this property is bolder on the microscale and nanoscale rather than the macroscale. This article is conducted to derive the governing equations of motion for a flexoelectric-multilayer structure and evaluate its vibrational behavior in response to various variable variations. The model includes a functionally graded (FG) porous core which is confined between two flexoelectricity-based face sheets at both sides and a Vlasov type of foundation. To gain the governing equations, the new type of power law for porous material beside Hamilton’s principle (HP) and modified couple stress theory (MCST) is implemented. Capturing the effects of length scale parameter, different materials, aspect and thickness ratios, porosity index, foundation parameter and boundary conditions (B.C.s) on the normalized natural frequency (NNF) of such a multilayer model based on the novel FG porous material formulation serve as the main novelty of this article. According to the results, it is revealed that the lateral ratio enhancement causes NNF and stiffness reduction in the model. Furthermore, among all considered B.C.s, simply support (SSSS) and clamped support (CCCC) refer to the lowest and the highest NNF respectively. The content of the recent article serves as a good source to provide a better understanding of the vibrational response of high stiffness-to-weight ratio structures to different variable variations. [ABSTRACT FROM AUTHOR]

Highly compliant structures such as microbeams can deform substantially in response to interactions between molecules adsorbed on their surface. To understand such systems and improve their detection signals, a mechano-electro-chemical coupling model for mechanical deformations of the microbeams immobilized single-stranded DNA (ssDNA) is established due to flexoelectricity. The governing equations and corresponding boundary conditions of ssDNA microbeams are derived by using the variational principle. The bending deformations of ssDNA microbeams (one for cantilever beam and another for simply supported beam) are derived. The electric potential in the regions inside and outside the ssDNA layer is obtained by linear Poisson-Boltzmann equation for different electrolyte solutions. The analytical expressions to quantify the beam deflection and the potential difference of ssDNA layer are presented. The theoretical predictions are compared with the experimental data to validate the applicability of the present model. Numerical results reveal that the solution types, thickness, and elastic modulus of substrate materials have an obvious influence on the deflections of ssDNA microbeams. Therefore, the present model can help to improve the reading of the bending deformation signal of the microbeam biosensors. [ABSTRACT FROM AUTHOR]

Flexible batteries are gaining momentum in several fields, including wearable medical devices and biomedical sensors, flexible displays, and smartwatches. These energy storage devices are subjected to electro-chemo-mechanical effects. Here, we present a theoretical framework that couples diffusion and electromechanical theory with flexoelectricity. As an example, we investigate the effect of flexoelectricity on the ionic conductivity in soft materials. Our analytical results for a thin film made of a soft material reveal that the ionic conductivity is significantly higher at the nanoscale and decreases exponentially to approach the bulk value with increasing film thickness. Furthermore, we find that flexoelectricity reduces the ionic conductivity dramatically at film thickness smaller than the length scale associated with flexoelectricity. This behavior is attributed to the opposite directions of polarization induced by flexoelectricity and the flow of ions driven by the chemical potential. These findings shed light on the interplay between flexoelectricity and diffusion which would be paramount in designing miniaturized energy storage devices. [ABSTRACT FROM AUTHOR]

With the rapid development and applications of micro/nano electromechanical systems in engineering fields, piezoelectric materials have been widely used in these systems due to their excellent physical and mechanical properties. Generally, these systems work in complex thermoelastic environments, to depict the thermal-electro-mechanical behaviors of piezoelectric material structures at microscale, the existing theories need to be developed due to the arising size-dependent effect, thermal relaxation effect and flexoelectric effect etc. To this end, the present work intends to generalize the size-dependent theories by considering the thermal-electro-mechanical coupling behaviors in piezoelectric material structures at microscale. Based on the nonlocal strain gradient theory and the generalized heat conduction model, a new model is established and the corresponding constitutive equations and the governing equations of equilibrium are derived. To illustrate the application of this new model, a piezoelectric micro-beam with flexoelectricity is considered and the closed form solutions of bending deflection and the free vibration response are obtained in terms of the boundary conditions. Based on the numerical results, the size-dependent effect, the piezo-flexoelectric effect and the thermal-lagging effect on the bending and vibration responses of the microbeam are examined and illustrated graphically. This work may be potentially helpful for the efficient, optimized and adaptive designs of the NEMS/MEMS devices. [ABSTRACT FROM AUTHOR]

Purpose: Many studies have found that when the piezoelectric devices are with the sizes of micrometer/nanoscale level, another significant electromechanical coupling characteristic, namely the flexoelectric effect, also plays important role in the performance of devices. In order to investigate the effect of flexoelectricity on the thickness-shear vibration, a mathematical model of a finite piezoelectric plate with slowly varying thickness and elliptical crosssection is presented with the consideration of flexoelectric effect in this paper. Methods: The proposed model uses the first order McLaughlin series to approximate the elliptical geometric function and the its first derivative is ignored since we assume that the plate thickness varies slowly along the length. The displacement function we adopted is assumed to be antisymmetric through the thickness due to the thicknessshear vibration mode. In addition, regarding to the thin and long flexoelectric plate, only the shear strain gradient through the thickness is considered in the mathematical model. The nonlinear governing equations, which incorporate geometric nonlinearity, are converted into differential equations as function of time by the Galerkin method. The method of multiple scales is employed to obtain the solution of nonlinear governing equation with the first order approximation. The obtained frequencies of thickness-shear mode are expressed in terms of vibration amplitude, eccentricity and flexoelectric coefficients. Results and Conclusion: The results show that the nonlinear thickness-shear vibration is size dependent and flexoelectric effect has significant influence on the nonlinear thickness-shear vibration frequencies of micro thin plates. The eccentricity and geometric nonlinearity also have great influence on the nondimensional fundamental frequencies. Thus, for accurate design of nano/micro scale high precision and high frequency flexoelectric/piezoelectric devices with elliptical cross-section, flexoelectricity must be taken into consideration. [ABSTRACT FROM AUTHOR]

Achieving renewable power for self‐powered electronic devices such as micro/nano‐electro‐mechanical systems is attractive in the fields of energy resources and environment, in this regard, dielectric material structures with excellent energy storage and electromechanical responses have stimulated a surge of research interest. However, in the micro/nanoscale filed, the existing thermo‐electromechanical continuum theories are not capable of fully and effectively characterizing the arising effects, for example, size‐dependent effect and heat lagging effect, etc. To further develop the existing theories, a theoretical prototype for piezo‐flexoelectricity by incorporating strain gradient effect and dual‐phase‐lag heat conduction model is proposed for the first time in this study, and then this proposed model is applied to investigating the energy harvesting performances of a piezo‐flexoelectric micro‐beam under piezoelectric‐thermoelastic coupling. In terms of the differential form of two‐phase elasticity and Hamilton's principle, the governing equations and the boundary conditions for the transverse vibration of a micro‐beam based on the Euler–Bernoulli beam model are derived. By means of the assumed mode method, the influences of the flexoelectricity, the load resistance and the strain gradient on voltage and output power are examined. From the obtained results, it can be aware of that this study not only presents the piezoelectric‐thermoelastic coupling effect of piezo‐flexoelectric configuration in microscale field, but also presents that a piezo‐flexoelectric system at microscale can generate higher power than that reported in previous works. It is hoped that this study could provide some guidelines in designing and optimizing dielectric energy harvesters of micro/nano electronic devices. [ABSTRACT FROM AUTHOR]

A new model of a bilayered transversely isotropic piezoelectric rectangular micro-plate with a distributed load is developed on the basis of Kirchhoff's plate theory and the extended linear piezoelectricity theory to characterize the piezoelectric micromachined ultrasonic transducer. The model takes into account both the surface effect and the flexoelectricity effect. The governing equation at the simply supported boundary condition is derived according to the variation principle. Based on the new model, the size dependent electromechanical coupling behaviors of the bilayered piezoelectric rectangular micro-plate are investigated. Considering the flexoelectric effect and surface effect synchronously, the numerical result indicates that the size dependence of the normalized central deflection decreases as the residual surface stress increases. For negative residual stress, the surface effect is the main influencing factor. While for positive residual stress, the surface effect dominates only when the ratio of thickness to length is smaller than about 25; otherwise, the flexoelectric effect will be more crucial. Moreover, if the thickness of the piezoelectric layer is less than about 40 nm, the electrical potential and polarization show a stronger size dependence. These results will be helpful to design and manufacture a piezoelectric micromachined ultrasonic transducer. [ABSTRACT FROM AUTHOR]

Using multiphysics simulations and experiments, we demonstrate that dielectric breakdown due to electric charge accumulation can lead to sufficient hotspot development leading to the initiation of chemical reactions in P(VDF-TrFE)/nAl films comprising a poly(vinylidene fluoride-co-trifluoroethylene) binder and nano-aluminum particles. The electric field (E-field) development in the material is driven by the flexoelectric and piezoelectric responses of the polymer binder to mechanical loading. A two-step sequential multi-timescale and multi-physics framework for explicit microscale computational simulations of experiments is developed and used. First, the mechanically driven E-field development is analyzed using a fully coupled mechanical–electrostatic model over the microsecond timescale. Subsequently, the transient dielectric breakdown process is analyzed using a thermal–electrodynamic model over the nanosecond timescale. The temperature field resulting from the breakdown is analyzed to establish the hotspot conditions for the onset of self-sustained chemical reactions. The results demonstrate that temperatures well above the ignition temperatures can be generated. Both experiments and analyses show that flexoelectricity plays a primary role and piezoelectricity plays a secondary role. In particular, the time to ignition and the time to pre-ignition reactions of poled films (possessing both piezoelectricity and flexoelectricity) are ∼10% shorter than those of unpoled films (possessing only flexoelectricity). [ABSTRACT FROM AUTHOR]

Significant flexoelectricity is expected to exist in materials with colossal permittivity. Here, we systematically studied the interplay of flexoelectricity and permittivity in CaCu3Ti4O12 (CCTO) ceramic by examining the thickness and electrode dependence of the flexoelectric coefficients over a wide range of temperatures. We found that an abnormal flexoelectric transition occurs at 95 °C. Below this critical temperature, the barrier layer mechanism dominates the significant flexoelectricity in CCTO ceramic, whereas above this critical temperature, the flexoelectric response mainly originates from the contributions of semiconducting grains and insulating grain boundaries. The observed flexoelectric transition is beneficial not only for developing new materials with high flexoelectric coefficients but also for understanding the colossal permittivity mechanism in CCTO ceramics. [ABSTRACT FROM AUTHOR]

Spontaneous polarization as large as ∼ 28 μ C / cm 2 was recently observed around the dislocation cores in non-polar SrTiO 3 bulk crystals, and its origin was attributed to the flexoelectric effect, i.e., polarization induced by strain gradients. However, the roles of flexoelectricity, relative to other electromechanical contributions, and the nature of dislocations, i.e., edge vs screw dislocations in the induced polarization, are not well understood. In this work, we study the role of flexoelectricity in inducing polarization around three types of dislocation cores in SrTiO 3 : b = a (100) edge dislocation, b = a (110) edge dislocation, and b = a (010) screw dislocation, where b is the Burgers vector. For the edge dislocations, polarization can be induced by electrostriction alone, while flexoelectricity is essential for stabilizing the symmetric polarization pattern. The shear component of the flexoelectric tensor has a dominant effect on the magnitude and spatial distribution of the flexoelectric polarization. In contrast, no polarization is induced around the b = a (010) screw dislocation through either electrostriction or flexoelectricity. Our findings provide an in-depth understanding of the role of flexoelectricity in inducing polarization around dislocation cores and offer insights into the defect engineering of dielectric/ferroelectric materials. [ABSTRACT FROM AUTHOR]

Global energy policies have urged power consumption reductions for electronic devices. However, this is inconsistent with the users' need for enhanced device performance. In this work, converse flexoelectricity is used to reduce power consumption while maintaining the superior display image quality of liquid crystal display (LCD). The first 1 Hz‐refresh‐rate and flicker‐free fringe‐field switching (FFS) LCD for a 13.4‐inch laptop are reported. A noticeable brightness deviation is inevitable when driving voltage polarity is altered during low‐frequency driving, which is caused by the flexoelectric effect and voltage‐holding failure. The nature of the converse flexoelectric contribution of mesogenic systems with the optimum flexoelectric coefficients and molecular polarity, and developed materials based on findings is investigated. The quality of static images and videos at 1‐ and 10 Hz refresh rates, respectively, is maintained, and the measured power consumptions are reduced by 83% and 47% compared to that at a 60 Hz refresh rate, and minimal flicker level is measured by −68 dB. It is believed that this will serve as a new paradigm for energy consumption in electronic devices with superior display performance of FFS LCDs. [ABSTRACT FROM AUTHOR]

In this paper, the electro-mechanical behavior of a flexoelectric nanobeam, considering the surface effect, is studied with an induced electric potential. Based on the modified strain gradient theory and Hamilton's variational principle, the governing differential equations of the nanobeam and the corresponding boundary conditions are obtained. The exact deflection solution of the flexoelectric nanobeam is derived from static bending. The relationship between the induced electric potential, including the surface effect, and the angle of the end of the cantilever beam is presented. Moreover, the characteristic equation of natural frequency is obtained by using a separate variable method under an open circuit with surface electrodes and an induced electric potential condition (OCI). The simulated results indicate the electro-mechanical response of the cantilever beam can be controlled by adjusting the flexoelectric coefficient, residual surface stress, and material length scale parameters of strain gradient theory. The free end of the beam with a platform phenomenon is also found by setting the appropriate parameters. The results also show that the residual surface stress and the ratio of beam thickness to material length scale parameters have a more significant effect on the effective frequency shift of the flexoelectric nanobeam. Therefore, considering the induced electric potential and surface effect is of great significance for the study of flexoelectric nanobeam sensors. [ABSTRACT FROM AUTHOR]

The separation of photogenerated electron–hole pairs is vitally important for photocatalysis, which can be effectively promoted by polarization field. However, it only manifests in piezoelectric/pyroelectric/ferroelectric materials that have a non-centrosymmetric structure. Here, we demonstrate that the polarization enhanced photocatalysis (with wide spectra from ultraviolet (UV) light to visible light) can be achieved in centrosymmetric semiconductors, such as δ-MnO2 and TiO2 nanosheets integrated nanoflowers, by using the strain-gradient-induced flexoelectric polarization that is always overlooked in polarization-enhanced catalysis. Under ultrasonic and illumination excitation, the organic pollutants (methylene blue (MB), etc.) can be effectively degraded within 30 min with excellent stability and repeatability. Compared with photocatalysis, the flexo-photocatalytic performance of above centrosymmetric semiconductors is substantially increased by 85%. Moreover, the factors related to flexo-photocatalysis such as material morphology, mechanical stimuli source, and adsorption are explored to deeply understand the mechanism of flexo-photocatalysis. This work opens up a way for high-performance photocatalysis in centrosymmetric semiconductors. [ABSTRACT FROM AUTHOR]

Purpose: This is the first investigation to employ an exact solution to explore the free vibrational behavior of beams under the impact of flexoelectricity and temperature when the beam is maintained by a viscoelastic foundation and subjected to axial compression. Methods: This study makes use of the analytical solution. Results: This study provided a solution in the form of an equation for estimating the natural frequency of oscillations of a beam resting on a viscoelastic substrate while accounting for the flexoelectric effect. This is the equation used to verify the accuracy of various approximation methods (FEM, isogeometric approach, etc.), and it is simple to use in reality to estimate the nanobeams' natural frequency quickly. In addition, the study findings provide a foundation for tackling more difficult issues and aid engineers in the design, production, and practical use of nanobeams. Conclusion: The article presented the exact solution to study the specific vibration of the beam, while it was resting on the viscoelastic medium while considering influences of the flexoelectricity. In this solution, effects of temperature and the influence of mechanical compressive force acting along the beam axis are both taken into account. The calculation formulae for beams are taken from the classical beam theory. When viscoelastic foundations are included, the particular vibrational response of the beam will comprise both real and imaginary parts. This discovery is quite fascinating, and it provides a platform for additional research into the mechanical reaction of beams when they are supported by a viscoelastic foundation. [ABSTRACT FROM AUTHOR]

Electroactivity of polymers used in energetic materials may result in charge separation that could result in safety concerns (unintentional ignition) or be exploited for multifunctional applications. We measured the flexoelectric properties of several polymers and energetic composites including poly(vinylidene fluoride-co-trifluoroethylene) [P(VDF-TrFE)], nanosized aluminum (nAl)/P(VDF-TrFE), poly(vinylidene fluoride-co-hexafluoropropylene) [P(VDF-HFP)], micron aluminum (μAl)/P(VDF-HFP), hydroxyl-terminated polybutadiene (HTPB), ammonium perchlorate (AP)/HTPB, μAl/AP/HTPB, polytetrafluoroethylene (PTFE), and polydimethylsiloxane (PDMS). The presence of flexoelectricity in PTFE (Teflon®) and the relatively high flexoelectric coefficient of P(VDF-HFP) (Viton®) measured in this work may help explain accidents involving the production and use of Magnesium-Teflon-Viton (MTV) that in many instances have been attributed to electro-static discharge. The addition of aluminum nanopowders to the P(VDF-TrFE) increased the flexoelectric coefficient by ∼30%. However, the addition of aluminum micrometer particles (10 wt. %) to P(VDF-HFP) decreased the effective flexoelectric coefficient, while an increase was observed when the aluminum loading was increased from 10 to 20 wt. %. The effective flexoelectric coefficient of HTPB and two propellant compositions (AP/HTPB and μAl/AP/HTPB) were measured to be in the same range as each other. The effect of particle addition (nAl, μAl, and AP) on flexoelectricity was different depending on the binder, further illustrating the complexity of flexoelectric properties in composite energetics. This may be somewhat explained by competing effects where particle additions (nAl, μAl, and AP) create additional strain gradients that contribute to flexoelectricity, but the particle additions also replace the mass of flexoelectric polymer binders (P(VDF-TrFE, P(VDF-HFP), and HTPB) with particles (nAl, μAl, and AP) that are less flexoelectric. [ABSTRACT FROM AUTHOR]

Donor doping can increase the dielectric constant of a material by several orders of magnitude due to induced space charge causing interfacial polarization. Giant dielectric properties and interface polarization are also both expected to greatly enhance the flexoelectric behavior of a material. In this work, a typical flexoelectric ceramic material, BaTiO3, was selected and donor doped using elemental Nb. Compared with the nominal BaTiO3 ceramic, the dielectric constant and flexoelectric coefficient of the Nb-doped BaTiO3 ceramics were significantly improved. The transverse flexoelectric coefficient of 0.3 mol. % Nb-BaTiO3 was found to increase to nearly 40 times the nominal value, reaching 387 μC/m. The results indicate that the giant dielectric response, and therefore the giant flexoelectric response, is the result of the combined effects of internal barrier-layer capacitance and surface barrier-layer capacitance. This study not only deepens the understanding of the semiconductor macro-dielectric effect and the flexoelectric mechanism caused by doping, but it also provides a feasible strategy for the design of giant dielectric/flexoelectric response materials and related devices with high dielectric constants and flexoelectric coefficients. [ABSTRACT FROM AUTHOR]

In this article, a geometrically exact flexoelectric beam model and its weak form quadrature element formulation are established. The coupling of strain gradient and electric field in flexoelectricity involving size effects is taken into account and the resulting C1 continuity requirements are satisfied. The formulation owns the advantage of high-order approximation to circumvent locking problems after discretization. Several numerical examples originating from benchmark nonlinear beam problems are studied to validate the feasibility of the formulation. Compared to existing flexoelectric beam models, the use of geometrically exact beam theory in the present model makes an innovative contribution to analyzing flexoelectric microbeams undergoing large displacements and rotations. This study is a step forward to better understandings of flexoelectric effects on microbeams under large deflections, while also providing a viable approach to predict their nonlinear behaviors in such conditions. [ABSTRACT FROM AUTHOR]

The phenomenon of flexoelectricity, wherein the generation of electrical polarization results from strain gradients, has garnered significant attention in electromechanics. This phenomenon holds immense potential for diverse applications in nanoelectromechanical systems. Doubly curved shells present a particularly compelling structure for implementing flexoelectric devices, representing the most general scenario. To comprehensively understand the mechanical behavior of nanoscale flexoelectric doubly curved shells, we present a novel analytical model in this study. Our model incorporates a reformulated strain gradient elasticity theory, a modified expression of electric enthalpy density considering Maxwell's self-field gradient, and a moderately thick shell configuration. To further elucidate this model, we establish complete formulations and analytical solutions of static bending and free vibration problems. This study reveals the crucial roles that various parameters such as principal curvature radii, length scale parameters, transverse structural sizes, and the Winkler–Pasternak foundation play in controlling the mechanical behavior of nanoscale flexoelectric doubly curved shells. These theoretical findings offer valuable insights into the design of nanodevices based on the direct flexoelectric effect. [ABSTRACT FROM AUTHOR]

The local scaling symmetry of the Lagrange density is exploited to investigate the myriad electro-mechanical coupling effects observed in the elastic dielectrics. In contrast to most known flexoelectricity theories, this approach has also explicated on the geometric underpinnings of the induced polarization and electric potential. Due to the inhomogeneous scaling of the metric, the invariance of the Lagrange density will be lost. To restore the gauge invariance of the Lagrange density, we introduce the notion of minimal replacement, viz., define a gauge covariant operator in place of the ordinary partial derivative. Minimal replacement introduces gauge compensating one form field. We relate the exact part of the gauge compensating one form field with electric potential. However, the gauge compensating one form field's anti-exact part is correlated with the electric polarization vector. We constructed different components of the gauge invariant energy density using the geometric objects like scale invariant gauge curvature. We appropriately contracted the skew symmetric gauge curvature tensor with the metric to construct the scale invariant energy associated with the gauge field. Finally, we derive the governing coupled equations employing Hamilton's principle. In order to assess how the theory performs, we carry out a few numerical simulations and validation. Model prediction of the pressurized flexoelectric disk with a central hole shows good agreement with the analytical solution. We also investigated flexoelectric cantilever beam subjected to a point load and different earthing conditions. Explorations of this kind of coupling may have notable implications in many industrial and laboratory applications. • Studied electro-mechanical coupling in dielectrics using the scaling symmetry. • Explored geometric underpinnings of induced polarization and electric potential. • Also recovered elastic Lagrangian assuming Euclidean manifold with gauge fixing. • Correspondence with Mindlin's polarization gradient theory for dielectrics. • This kind of coupling may have notable implications in many applications. [ABSTRACT FROM AUTHOR]

Within the framework of the consistent couple stress theory (CCST), we develop a Hermitian Cn (n = 1, 2) finite cylindrical layer method (FCLM) for carrying out the three‐dimensional (3D) analysis of the size‐dependent buckling and free vibration behaviors of simply supported, functionally graded (FG) piezoelectric cylindrical sandwich microshells. The microshells of interest are placed under closed‐circuit surface conditions and subjected to axial compression and electric voltages. We derive a 3D weak formulation based on Hamilton's principle for this study. In the resulting formulation, the microshell is artificially divided into nl microlayers, with the elastic displacement components and the electric potential selected as the primary variables. By incorporating a layer‐wise kinematic model into our weak formulation, we develop a Hermitian Cn FCLM, which can be used for analyzing FG piezoelectric cylindrical sandwich microshells. Each primary variable is expanded as a double Fourier series in the in‐surface domain and is interpolated in the thickness direction using Hermitian Cn polynomials. The accuracy and the convergence rate of our Hermitian Cn FCLMs are validated by comparing the solutions they produce for FG piezoelectric cylindrical macroshells and FG elastic cylindrical microshells with the relevant exact and approximate 3D solutions which have been reported in the literature. The material length scale parameter of our FCLMs is set at zero in the comparison made with the FG piezoelectric macroshells. In contrast, the piezoelectric and flexoelectric effects are ignored in the comparison made with the FG elastic microshells. The impact of some essential factors on the critical load, critical voltage, and natural frequency of simply supported FG piezoelectric cylindrical sandwich microshells is assessed. The important factors are identified as piezoelectricity, flexoelectricity, the material length scale parameter, the inhomogeneity index, the radius‐to‐thickness ratio, the length‐to‐radius ratio, and the magnitude of the applied voltage and the applied load. [ABSTRACT FROM AUTHOR]