Two-dimensional Behavior Research Articles

Formulation of an efficient shear-flexure interaction model for planar reinforced concrete walls

A research was conducted to develop a macroscopic modeling approach that integrates axial, flexure, and shear interaction under cyclic loading conditions to obtain reliable predictions of the nonlinear response of reinforced concrete (RC) structural walls. The model, named as Efficient-Shear-Flexure-Interaction (E-SFI), is intended to provide accurate results for squat, medium-rise, and slender planar walls, with a computationally efficient formulation that can be used under generalized conditions. The E-SFI model, which is based on the Shear-Flexure Interaction Multiple-Vertical-Line-Element-Model (SFI-MVLEM), incorporates a two-dimensional RC panel behavior described with a fixed-crack-angle approach. The novel formulation removes the internal degree of freedom per RC panel element of the SFI-MVLEM by incorporating a calibrated expression to compute the horizontal normal strain (εx), and therefore removing the assumption of zero resultant horizontal stress (σx=0) to increase the range of applicability of the model. Three RC wall specimen tests were selected to calibrate and validate the E-SFI model, as well as to prove the model efficiency, including a slender wall (shear span-to-depth ratio of 3.0), a medium-rise wall (shear span-to-depth ratio of 1.5), and a squat wall (shear span-to-depth ratio of 0.63). Analytical model results reveal the ability of the model to accurately reproduce the hysteretic response for all considered cases, with an important reduction in the runtime and an improved current tangent convergence rate compared with the SFI-MVLEM.

Controlling the anisotropy of a van der Waals antiferromagnet with light.

Van der Waals magnets provide an ideal playground to explore the fundamentals of low-dimensional magnetism and open opportunities for ultrathin spin-processing devices. The Mermin-Wagner theorem dictates that as in reduced dimensions isotropic spin interactions cannot retain long-range correlations, the long-range spin order is stabilized by magnetic anisotropy. Here, using ultrashort pulses of light, we control magnetic anisotropy in the two-dimensional van der Waals antiferromagnet NiPS3 Tuning the photon energy in resonance with an orbital transition between crystal field split levels of the nickel ions, we demonstrate the selective activation of a subterahertz magnon mode with markedly two-dimensional behavior. The pump polarization control of the magnon amplitude confirms that the activation is governed by the photoinduced magnetic anisotropy axis emerging in response to photoexcitation of ground state electrons to states with a lower orbital symmetry. Our results establish pumping of orbital resonances as a promising route for manipulating magnetic order in low-dimensional (anti)ferromagnets.

Supramolecular Architecturing of Quantum Box Arrays from Functionalized Porphyrins and Exploring Their Quantum States

Our research is based on the analysis of structure-function correlations for a broad spectrum of functionalized porphyrins and phthalocyanines in their propensity to create increasingly complex and functional on surface architectures. [1] Thereby we use porphyrin and phthalocyanine functionalization and synthesis, ‘by design’, or ‘on-surface-supramolecular chemistry’ to create structures with specific physical properties towards e.g. future quantum devices.By using a fluoro-bi-phenyl functionalized porphyrin to create a stable, porous organic network, we manufacture arrays of identical quantum boxes. These boxes contain a specific quantum well state arising from the confinement of the electronic (Shockley) surface state within the pores. We show that the electronic quantum box state as well as the interbox coupling can be modified locally by adsorbates, e.g. C60, interacting with the barrier. In view of the wealth of available and differently acting adsorbates, this approach allows for the engineering of quantum states in on-surface network architectures. Thus, by their dance on the nose of the ‘devil’ [3], porphyrins and phthalocyanines justify the emergence of ‘on-surface supramolecular physics’ as reflected by ongoing and future research work.Fig. 1) Adsorbate-induced modification of the confining barriers in a quantum box array formed by a functionalized fluoro-bi-phenyl porphyrin on Ag(111). (Taken from [2])[1] N. Wintjes et al. Two-dimensional phase behavior of a bimolecular porphyrin system at the solid-vacuum interface, J. Am. Chem. Soc. 2010, 132, 21, 7306-7311; N. Wintjes et al., Supramolecular synthons on surfaces: Controlling dimensionality and periodicity of tetraarylporphyrin assemblies by the interplay of cyano and alkoxy substituents Chem. Eur. J. 2008, 14, 5794.[2] S. Nowakowska et al., Adsorbate-induced modification of the confining barriers in a quantum box array, ACS Nano 2018, 12, 768.[3] W. Pauli “God made the bulk; surfaces were invented by the devil.” Quoted, e.g. in the Preface of A. Zangwill, Physics at Surfaces, Cambridge University Press, 1988. ISBN 978-0-521-34752-5 Figure 1

3D to 2D Magnetic Ordering of Fe3+ Oxides Induced by Their Layered Perovskite Structure

The antiferromagnetic behavior of Fe3+ oxides of composition RE1.2Ba1.2Ca0.6Fe3O8, RE2.2Ba3.2Ca2.6Fe8O21, and REBa2Ca2Fe5O13 (RE = Gd, Tb) is highly influenced by the type of oxygen polyhedron around the Fe3+ cations and their ordering, which is coupled with the layered RE/Ba/Ca arrangement within the perovskite-related structure. Determination of the magnetic structures reveals different magnetic moments associated with Fe3+ spins in the different oxygen polyhedra (octahedron, tetrahedron, and square pyramid). The structural aspects impact on the strength of the Fe-O-Fe superexchange interactions and, therefore, on the Néel temperature (TN) of the compounds. The oxides present an interesting transition from three-dimensional (3D) to two-dimensional (2D) magnetic behavior above TN. The 2D magnetic interactions are stronger within the FeO6 octahedra layers than in the FeO4 tetrahedra layers.

Using Oil Spill Modeling in Oil Spill Exercises and Drills

ABSTRACT Oil spill trajectory and fate modeling was used in inland response Full Scale Exercises including the Enbridge Des Plains River (fall 2018) and Wisconsin River (fall 2019). The Spill Impact Model Application Package (SIMAP) was used to predict the three-dimensional movement (i.e. trajectory) and behavior (i.e. fate) of a hypothetical release of oil using site-specific environmental and geographic conditions (including seasonal and hydrographic information) for the date of the exercise. The RPS OILMAPLand model was also used to predict the two-dimensional movement and behavior of the oil over the land surface, before it was predicted to enter the waterway. The oil spill modeling evaluated the spatial extent, timing, and magnitude of hydrocarbon contamination at downstream locations including thicknesses of floating surface oil and the mass of oil on shorelines and sediments. The assessments included the potential for released oil to move over the land surface, before entering the waterway, as well as becoming entrained in the water column as a result of surface floating oil passing over local features such as locks and dams. The results were presented at two separate exercise planning session and the full scale exercise as static images, GIS shape files, and videos. Results were also included in the COP for the exercise itself, with predicted results provided at hourly intervals for several days.

Theoretical Prediction of Two-Dimensional Materials, Behavior, and Properties.

Predictive modeling of two-dimensional (2D) materials is at the crossroad of two current rapidly growing interests: 2D materials per se, massively sought after and explored in experimental laboratories, and materials theoretical-computational models in general, flourishing on a fertile mix of condensed-matter physics and chemistry with advancing computational technology. Here the general methods and specific techniques of modeling are briefly overviewed, along with a somewhat philosophical assessment of what "prediction" is, followed by selected practical examples for 2D materials, from structures and properties, to device functionalities and synthetic routes for their making. We conclude with a brief sketch-outlook of future developments.

Some Aspects of Seismic Soil–Structure Interaction of Lifeline Structures

Historically, underground structures were considered to be less vulnerable to earthquakes. However, some of the recent earthquakes have demonstrated that underground structures too can suffer severe damages, especially when these are located in the vicinity of causative faults. In case of a shallow underground tunnel excavated in an urban area, design and construction is demanding due to interaction between the tunnels and the overlying pre-existing structures. Selection of a realistic ground motion of an earthquake plays a crucial role in the seismic analysis and design of underground structures. In the absence of realistic earthquake data for the given area, it becomes essential to generate artificial or synthetic time history compatible with the largest earthquake expected in that area. Depending upon the situation, it may become necessary to consider three-dimensional aspects of the problem rather than depending on only a two-dimensional behavior. Strong ground shaking can as well cause loss of strength in saturated cohesion less soils resulting in liquefaction. Liquefaction can cause the ground surrounding the tunnels to shift, with potentially severe consequences. An attempt has been made in this paper to discuss various analysis and design considerations of underground lifeline structures and then look in to the aspects of stability of metro underground tunnels of Delhi city on basis of response spectra compatible time histories of 1999 Chamoli earthquake of Uttarakhand, actual three-dimensional analyses, and some liquefaction studies.

Three-dimensional similarity law for elastic wave propagation in CFRP with concentrically curved fibers

The wave front of an elastic wave in carbon fiber reinforced plastic (CFRP) with curved fibers is distorted owing to the fiber curvature effect. The similarity law to estimate the two-dimensional (2D) elastic bulk wave behavior in CFRP with concentrically curved fibers was presented in our previous study. In this research, the three-dimensional (3D) propagation of bulk waves in CFRPs with concentrically curved fibers was examined. Using 3D-printed CFRP specimens, the 3D propagation behavior of the quasi-longitudinal wave emitted from the point source was visualized virtually. Considering these results, the similarity law was expanded to the case of the 3D propagation behaviors of the bulk waves. Mathematical discussion based on the governing equations shows that the expansion parameters need to be equal in both the thickness and the radial directions to satisfy the similarity law in the 3D case, in addition to the conditions required to satisfy the 2D similarity law. By conducting the experimental visualization, it was demonstrated that the shape changes of quasi-longitudinal waves follow the similarity law, which was eventually verified for 3D propagation.

Layered Hexagonal Perovskite Oxides 21R Ba7Fe5Ge2O20 and 12H Ba6Fe3Ge3O17.

Two hexagonal-perovskite-structure oxides, 21R Ba7Fe5Ge2O20 and 12H Ba6Fe3Ge3O17, were obtained by synthesis with a high-pressure and high-temperature technique. The Fe-containing hexagonal-perovskite-structure units are sandwiched by nonmagnetic GeO4 tetrahedral layers in the structures, and thus both compounds show two-dimensional ferrimagnetic behaviors due to intra- and interunit magnetic interactions. 21R Ba7Fe5Ge2O20 has the ionic formula Ba7Fe123+Fe24+Fe324+Ge424+O20 at room temperature, and unusually high valence Fe4+ in the trimers undergoes charge disproportionation, Fe24+ + 2Fe34+ → Fe2(4+2δ)+ + 2Fe3(4-δ)+, at low temperatures. In contrast, 12H Ba6Fe3Ge3O17 with ionic formula Ba6Fe123+(Fe20.54+Ge20.54+)2Ge324+O17 does not show a charge transition.

Temperature-sensitive soft microgels at interfaces: air-water versus oil-water.

The formation of smart emulsions or foams whose stability can be controlled on-demand by switching external parameters is of great interest for basic research and applications. An emerging group of smart stabilizers are microgels, which are nano- and micro-sized, three-dimensional polymer networks that are swollen by a good solvent. In the last decades, the influence of various external stimuli on the two-dimensional phase behavior of microgels at air- and oil-water interfaces has been studied. However, the impact of the top-phase itself has been barely considered. Here, we present data that directly address the influence of the top-phase on the microgel properties at interfaces. The dimensions of pNIPAM microgels are measured after deposition from two interfaces, i.e., air- and decane-water. While the total in-plane size of the microgel increases with increasing interfacial tension, the portions or fractions of the microgels situated in the aqueous phase are not affected. We correlate the area microgels occupy to the surface tensions of the interfaces, which allows to estimate an elastic modulus. In comparison to nanoindentation measurements, we observe a larger elastic modulus for the microgels. By combining compression, deposition, and visualization, we show that the two-dimensional phase behavior of the microgel monolayers is not altered, although the microgels have a larger total in-plane size at higher interfacial tension.

Fermi surface and 2D-Electron momentum density of covellite mineral

Abstract We report the Fermi surface and the two-dimensional electron momentum density (2D-EMD) of Covellite mineral applying LDA, GGA and GGA + U schemes in the LAPW method. Covellite, CuS owes hcp structure. Though it is found in pure form, traces of other crystals remain during exploration. The 2D-EMD studied using 2D-ACAR is highly sensitive to impurities. Moreover, shape of Fermi surface can also be obtained by 2D-EMD. Therefore, in this work the 2D-EMD is calculated and the shape of the Fermi surface of Covellite is presented. The Fermi surface owes corrugated cylindrical shape coaxial to the Г-A direction. This marks very low conductivity along the cylinder axis at low temperatures suggesting a two-dimensional behaviour. The elliptical shape due to corrugated cylinder is also clearly visible in the 2D-EMD.

Cracking analysis of plane stress reinforced concrete structures

This article presents the numerical analysis’ results of reinforced concrete elements subjected to plane stress in early cracking stage. The elements are modelled by a concrete two-dimensional matrix, discrete reinforcement bars and bond-slip elements. The aim of this study is to investigate the behaviour of RC structures before and after the formation of the first cracks to understand the influence on the crack spacing and width of bar orientation with respect to the crack direction, bar spacing and diameter and presence of shear stresses on the crack. Discrete crack non-linear analysis of elements with reinforcement both orthogonal and skew to the crack directions are performed. The interaction between concrete and steel is ensured by a non-linear bond slip law at the interfaces between the two materials. The crack spacing obtained numerically are compared with the ones calculated using different design codes. The analysis of models with different reinforcement geometries allows individuating and discussing the main factors governing two-dimensional plane stress concrete cracking behaviour.

Langmuir-Blodgett Films of Chiral Perfluorinated Gelators: Effects of Chirality and Chain Length on Two-Dimensional Behavior

Abstract Surface pressure versus molecular area (π-A) curves were measured on pure water as a subphase for a series of N,N′-diperfluoroalkanoyl-1,2(R,R)-diaminocyclohexanes. A molecule is denoted as RR-CFn, where n is the number of carbon atoms in a perfluoroalkanoyl chain (or n − 2 = the number of difluoromethylene units). The chain length was varied for n = 4, 5, 6, 8, 9, and 10. The results for n = 7 were reported previously. The effects of chain length and optical purity on film formation were investigated. The surface morphology of a film deposited onto a hydrophilic glass plate was observed using an atomic force microscope (AFM). For n = 4, 5, and 6, the floating films were already multilayered before compression and the deposited films were composed of rectangular or rod-like aggregates. For n = 7, 8, 9, and 10, monolayered films were formed and underwent structural transformation upon compression. From the AFM images, the films deposited after the transformation were composed of fiber-like aggregates. For a racemic mixture, no monolayer film was formed, and the film transfer was impossible irrespective of the chain length. p-Polarized infrared multiple angle incidence resolution spectrometry (pMAIRS) measurements were carried out on a film of RR-CF8 deposited onto a silicon wafer to determine the orientation of the composite molecules. The results were compared with the monolayer behavior reported for a compound having a single perfluoroalkyl chain. The relation to their gelation behavior is also discussed.

A shock-stable modification of the HLLC Riemann solver with reduced numerical dissipation

The purpose of this paper is twofold. First, the application of high-order methods in combination with the popular HLLC Riemann solver demonstrates that the grid-aligned shock instability can strongly affect simulation results when the grid resolution is increased. Beyond the well-documented two-dimensional behavior, the problem is particularly troublesome with three-dimensional simulations. Hence, there is a need for shock-stable modifications of HLLC-type solvers for high-speed flow simulations.Second, the paper provides a stabilization of the popular HLLC flux based on a recently proposed mechanism for grid aligned-shock instabilities Fleischmann et al. (2020) [8]. The instability was found to be triggered by an inappropriate scaling of acoustic and advection dissipation for local low Mach numbers. These low Mach numbers occur during the calculation of fluxes in transverse direction of the shock propagation, where the local velocity component vanishes. A centralized formulation of the HLLC flux is provided for this purpose, which allows for a simple reduction of nonlinear signal speeds. In contrast to other shock-stable versions of the HLLC flux, the resulting HLLC-LM flux reduces the inherent numerical dissipation of the scheme.The robustness of the proposed scheme is tested for a comprehensive range of cases involving strong shock waves. Three-dimensional single- and multi-component simulations are performed with high-order methods to demonstrate that the HLLC-LM flux also copes with latest challenges of compressible high-speed computational fluid dynamics.

İKİ BOYUTLU FONKSİYONEL KADEMELENDİRİLMİŞ PLAKALARIN YAPAY SİNİR AĞI ÖĞRENME ALGORİTMALARI İLE ISIL GERİLME MODELLEMESİ

Belirli bir hacimsel dağılım fonksiyonuna göre oluşturulan fonksiyonel kademelendirilmiş malzemeler (FKM) günümüz yüksek sıcaklık uygulamalarına dayanımlı malzeme üretiminde önemli bir yere sahiptir. Amaç fonksiyonuna bağlı olarak FKM’de maksimum çalışma performansı, yapısal varyasyonlar ve emniyetli gerilme değerleri gibi önemli özelliklerin sağlanabilmesi için hacimsel dağılımın belirlenmesi çok önemlidir. Hacimsel dağılımın belirlenmesi ve uygun hacimsel dağılımın test edilebilmesi için nümerik analiz yöntemleri kullanılmaktadır. Bu çalışmada, sonlu farklar metodu ile iki boyutlu fonksiyonel kademelendirilmiş plakaların ısı akısı tesirinde ısıl-mekanik davranışının tespit edilmesinde önemli bir parametre olan eşdeğer gerilme seviyeleri için malzemenin hacimsel dağılımına bağlı modeller oluşturulmuştur. Bu modeller yapay sinir ağında iki farklı eğitim algoritması ile elde edilmiştir. Bu eğitim algoritmaları Levenberg-Marquart ve Geriye Yayılım algoritmasıdır. FKM’lerin hacimsel dağılımını belirleyen kompozisyonel gradyant üst değerleri n ve m’dir. Sonlu farklar metodu ile sayısal analizde n ve m değerlerini belirlemek 1800 s iken Yapay Sinir ağı modeli ile 900 s olup verimlilik önemli derecede artmaktadır. Kullanılan eğitim algoritmalarının iş-zaman-performans değerleri açısından ileride yapılacak bilimsel çalışmalar için fikir verici nitelikte olması önemlidir. Önerilen eğitimli modeller henüz seri üretimi gerçekleştirilemeyen FKM için hem üretimde hem de yapılacak teorik çalışmalarda optimum hacimsel dağılıma ulaşmada yol gösterici olacaktır.

Unconventional superconductivity in CuxBi2Se3 from magnetic susceptibility and electrical transport

Although the Cu doped Bi2Se3 topological insulator was discovered and intensively studied for almost a decade, its electrical and magnetic properties in normal state, and the mechanism of ‘high-Tc’ superconductivity regarding the relatively low-carrier density are still not addressed yet. In this work, we report a systematic investigation of magnetic susceptibility, critical fields, and electrical transport on the nominal Cu0.20Bi2Se3 single crystals with = 4.18 K, the highest so far. The composition analysis yields the Cu stoichiometry of x = 0.09(1). The magnetic susceptibility shows considerable anisotropy and an obvious kink at around 96 K was observed in the magnetic susceptibility for H∥c, which indicates a charge density anomaly. The electrical transport measurements indicate the two-dimensional (2D) Fermi liquid behavior at low temperatures with a high Kadowaki–Woods ratio, A/γ2 = 30.3a0. The lower critical field at 0 K limit was extracted to be 6.0 Oe for H∥ab. In the clean limit, the ratio of energy gap to Tc was determined to be Δ0/kBTc = 2.029 ± 0.124 exceeding the standard BCS value 1.764, suggesting Cu0.09Bi2Se3 is a strong-coupling superconductor. The in-plane penetration depth at 0 K was calculated to be 1541.57 nm, resulting in an unprecedented high ratio of Tc/λ−2(0) ≅ 9.86. Moreover, the ratio of Tc to Fermi temperature is estimated to be = 0.034. Both ratios fall into the region of unconventional superconductivity according to Uemura’s regime, supporting the unconventional superconducting mechanism in CuxBi2Se3. Finally, the enhanced Tc value higher than 4 K is proposed to arise from the increased density of states at Fermi energy and strong electron–phonon interaction induced by the charge density instability.

On-Surface Supramolecular Chemistry with Porphyrins and Phthalocyanines: An Architectural Concept Leading to Engineered Quantum-Functional Nanostructures

The availability of porphyrin and phthalocyanine functionalization and synthesis, ‘by design’, at the time when sub-molecular resolution imaging at surfaces came into action boosted the emergence of ‘On-Surface-Supramolecular Chemistry’. [1] It is instructive to compare supra-molecular binding motifs in the fluid phase with the corresponding or modified analogues found after assembly on surfaces. In some similarity to chemistry and coordination chemistry translating into the corresponding ‘on-surface’ sciences, there are many factors complicating the on-surface assembly and architecture. Examples are provided by the modified dimensionality and kinetics, the site-specific conformational flexure and chirality and the close to irreversible adsorption in the strong surface potential in absence of a solvent which may complicate the on-surface assembly and architecture. Further, porphyrins and phthalocyanines interact with the substrate by different, competitive interactions, i.e. via (1) the central metal atom and (2) the porphyrin pi-system in addition to (3) the functional side groups. Binary and ternary assemblies of porphyrins and other functional components make it possible to create a wealth of ‘phases’, i.e. different complex supramolecular architectures with interesting physical properties to explore and exploit.We analyze structure-function correlations for a broad spectrum of functionalized porphyrins and phthalocyanines in their propensity to create structures of increasing complexity and functionality, on surfaces, also for future quantum technology. Notably, the large delocalized electronic system of the porphyrins and their easy-to-replace central atom further extend the achievable range of architectures and functions. Thus, by their dance on the nose of the ‘devil’ [3], porphyrins and phthalocyanines justify the emergence of ‘on-surface supramolecular physics’ as reflected by ongoing and future research work.[1] N. Wintjes et al. Two-dimensional phase behavior of a bimolecular porphyrin system at the solid-vacuum interface, J. Am. Chem. Soc. 2010, 132, 21, 7306-7311; N. Wintjes et al., Supramolecular synthons on surfaces: Controlling dimensionality and periodicity of tetraarylporphyrin assemblies by the interplay of cyano and alkoxy substituents Chem. Eur. J. 2008, 14, 5794.[2] S. Nowakowska et al., Adsorbate-induced modification of the confining barriers in a quantum box array, ACS Nano 2018, 12, 768.[3] W. Pauli “God made the bulk; surfaces were invented by the devil.” Quoted, e.g. in the Preface of A. Zangwill, Physics at Surfaces, Cambridge University Press, 1988. ISBN 978-0-521-34752-5Fig. 1) Adsorbate-induced modification of the confining barriers in a quantum box array formed by a functionalized fluoro-bi-phenyl porphyrin on Ag(111). (Taken from [2]) Figure 1

Two-Dimensional Mechanical Behavior Analysis of Multilayered Solids Subjected to Surface Contact Loading Based on a Semi-Analytical Method

In this paper, a two-dimensional semi-analytical method is developed for the mechanical behavior analysis of multilayered solids subjected to surface contact loading, which is indispensable for realizing an optimized tribological performance from the mechanical behavior point of view. Firstly, the explicit analytical frequency response functions of the multilayered solid are derived in a recursive form by analytically solving a system of linear equations established according to the boundary conditions and the interface continuous conditions. Then, the two-dimensional elastic field solution in the subsurface of multilayered solids in the space domain is converted from its corresponding frequency response functions by employing a numerical conversion method based on the inverse fast Fourier transformation. The present method is validated by comparing with the solution given by other methods. Lastly, the stress analysis of multilayered coatings with various structure layouts and various layer number of the multilayers were performed with the present method.

Abrupt Transition between Three-Dimensional and Two-Dimensional Quantum Turbulence.

We present numerical evidence of a critical-like transition in an out-of-equilibrium mean-field description of a quantum system. By numerically solving the Gross-Pitaevskii equation we show that quantum turbulence displays an abrupt change between three-dimensional (3D) and two-dimensional (2D) behavior. The transition is observed both in quasi-2D flows in cubic domains (controlled by the amplitude of a 3D perturbation to the flow), as well as in flows in thin domains (controlled by the domain aspect ratio) in a configuration that mimics systems realized in laboratory experiments. In one regime the system displays a transfer of the energy towards smaller scales, while in the other the system displays a transfer of the energy towards larger scales and a coherent self-organization of the quantized vortices.

Two-dimensional bending behavior of the three-layered shear deformable nanoshells: Electro-elastic size-dependent

First-order shear deformation theory and nonlocal piezoelasticity relations are employed in this work for size-dependent electro-elastic bending analysis of three-layered piezoelectric nanoshells. The three-layered nanoshell includes a nano core and two piezo-magnetic face-sheets as sensor and actuator. A two-dimensional formulation along the axial and radial directions is presented based on first-order shear deformation theory to account shear strains and stresses especially at boundaries. Principle of virtual work is used to derive the governing equations of electro-elastic bending for a size-dependent piezoelectric nanoshell. The numerical results are presented based on analytical approach to investigate the influence of significant parameters such as nonlocal parameter, two parameters of Pasternak’s foundation and initial electric potential on the electro-elastic bending results.