External Magnetic Potential Research Articles

Free Vibration of the Magneto-Electro-Elastic Plates Resting on Elastic Foundation Using the Refined Plate Theory with Two Variables

The objective of this article is to investigate the free vibration analyses of the functionally graded magneto-electro-elastic (FG MEE) plates supported by an elastic foundation using the refined plate theory (RPT) with two variables. The elastic foundation is modeled utilizing the Winkler-Pasternak theory. The power-law model is employed to characterize the graded material properties of the FG MEE plates. According to the RPT and Hamilton's principle, the governing equations of the FG MEE plate are derived. The displacement fields and electric and magnetic potentials are approximated using the Non-Uniform Rational B-Splines (NURBS) basic functions of the isogeometric approach (IGA). The proposed model’s advantages and accuracy are demonstrated by comparing the obtained results with those reported in the existing literature. The study comprehensively examines and discusses the impact of several parameters, including the power index, initial external magnetic potential and electric voltage, and the geometry, on the vibration frequency of the FG MEE plates. The numerical findings indicate that an increase in the power index leads to a decrease in the frequency of the FG MEE plates. Besides, the stiffness of FG MEE plates decreases with an increase in the initial external electric voltage, whereas it increases with an increase in the initial external magnetic potential. This article presents valuable perspectives on examining vibration analysis of the FG MEE plates, which can inform the design of innovative materials and structures with customized properties.

Thermomechanical Response of Smart Magneto-Electro-Elastic FGM Nanosensor Beams with Intended Porosity

AbstractThis study investigates the behavior of free vibrations in a variety of porous functionally graded nanobeams composed of ferroelectric barium-titanate (BaTiO3) and magnetostrictive cobalt-ferrite (CoFe2O4). There are four different models of porous nanobeams: the uniform porosity model (UPM), the symmetric porosity model (SPM), the porosity concentrated in the bottom region model (BPM), and the porosity concentrated in the top region model (TPM). The nanobeam constitutive equation calculates strains based on various factors, including classical mechanical stress, thermal expansion, magnetostrictive and electroelastic properties, and nonlocal elasticity. The study investigated the effects of various factors on the free vibration of nanobeams, including thermal stress, thermo-magneto-electroelastic coupling, electric and magnetic field potential, nonlocal features, porosity models, and changes in porosity volume. The temperature-dependent mechanical properties of BaTiO3 and CoFe2O4 have been recently explored in the literature for the first time. The dynamics of nanosensor beams are greatly influenced by temperature-dependent characteristics. As the ratios of CoFe2O4 and BaTiO3 in the nanobeam decrease, the dimensionless frequencies decrease and increase, respectively, based on the material grading index. The dimensionless frequencies were influenced by the nonlocal parameter, external electric potential, and temperature, causing them to rise. On the other hand, the slenderness ratio and external magnetic potential caused the frequencies to drop. The porosity volume ratio has different effects on frequencies depending on the porosity model.

The Klein-Gordon equation in the cosmic string and rainbow gravity spacetime under the influence of an external magnetic field and Coulomb potential for PDM particles

Abstract The Klein-Gordon equation in the cosmic string and rainbow gravity spacetime under the influence of an external magnetic field and Coulomb potential for PDM particles is investigated using the NU method. From solution of the Klein-Gordon system with PDM particles, the energy levels are obtained. The energy levels are numerically calculated as a function of rainbow gravity parameters, as a function of magnetic field parameters, as a function of position-dependent mass parameters. We apply some rainbow functions and the results show that the negative and positive energy levels for rainbow function 1 are symmetrical compared to rainbow function 2.

The effect of the foam structure on the thermomechanical vibration response of smart sandwich nanoplates

This study aimed to model the thermomechanical vibration behavior of sandwich nanoplates consisting of a foam or solid core layer and smart surface layers. The sandwich plate’s mechanical behavior is analyzed using the nano-size effect, nonlocal strain gradient theory, and sinusoidal higher-order plate theory. Hamilton’s principle is employed to derive the equations of motion, which are subsequently solved using the Navier method. The impacts of temperature, foam structure, foam void ratio, smart materials, and the external electric and magnetic potentials applied to the surface layers on the thermomechanical behavior of a sandwich plate have been analyzed.

Nonlinear deformation analysis of magneto-electro-elastic nanobeams resting on elastic foundation by using nonlocal modified couple stress theory

This article investigates a comprehensive analysis of the bending problem pertaining to magneto-electro-elastic (MEE) nanobeams on the Winkler-Pasternak foundation. Taken into account the influence of size effects of nano structures, a non-classical mechanical model is established to describe the bending behavior for MEE nanobeams. In this research, the nonlocal modified couple stress theory is utilized to accurately capture both the softening and hardening effects resulting from the size effect. The theory of third-order shear deformation and von Karman geometric nonlinear theory are employed in conjunction with the introduction of Maxwell's equation to develop the model of MEE nanobeam, the control equation of the nonlinear bending problem of MEE nanobeams is obtained by Minimum potential energy principle, and it is solved by Galerkin method. Finally, this study provides a detailed discussion on the influences of the ratio of nonlocal parameters and length-scale parameters, Winkler-Pasternak coefficient, external magnetic potential, external voltage and span-thickness ratio on nonlinear deflection of nanobeam.

Isogeometric vibration of the magneto-electro-elastic sandwich plate with functionally graded carbon nanotube reinforced composite core

This paper investigates the free vibration analysis of the magneto-electro-elastic (MEE) sandwich plates with functionally graded carbon nanotube reinforced composite (FG-CNTRC) core using isogeometric approach (IGA). The sandwich plate is composed of the homogeneous MEE face sheets and FG-CNTRC core with four types of carbon nanotubes (CNTs) distribution, including CNT-UD, CNT-O, CNT-V and CNT-X. The external electric voltage and magnetic potential are applied in the top and bottom layers of the MEE sandwich plate. Employing the refined plate theory (RPT) and Hamilton principle, the governing equation for free vibration of the MEE sandwich plate is derived. The IGA employs Non-Uniform Rational B-Splines (NURBS) basic functions to approximate the displacement fields and the magnetic and electric potentials in the RPT model. The study examines and discusses the impact of different factors on the frequency of the MEE sandwich plate, including parameters like CNTs distributions, CNTs volume fraction, external electric voltage and magnetic potential, and the geometrical parameter of the plate. This research has revealed several important discoveries regarding the fabrication of MEE sandwich structures.

Nonlinear dynamic modeling of geometrically imperfect magneto-electro-elastic nanobeam made of functionally graded material

Smart magneto-electro-elastic (MEE) structures with high performance are highly valuable across a vast range of aerospace and industrial applications like smart sensing, space robotics, energy harvesting and biomedical research. This work focuses on the nonlinear dynamic response of a functionally graded MEE nanobeam. Initial geometric imperfection including global and localized imperfection modes is considered to reflect the effect of undesired error during nanofabrication process. Such geometric imperfection is involved by the product of hyperbolic and trigonometric functions. The nonlinear dynamic model for the FG-MEE nanobeam is established according to Hamilton’s principle and nonlocal constitutive relation to derive its governing equations of motion. The interaction between the imperfect FG nanobeam and its multi-physical stimulus is numerically solved via differential quadrature method. After validating the proposed model with existing works, the parametric study is conducted to fully reveal the coupling effect of geometric imperfection and numerous parameters including imperfection mode and amplitude, external electric and magnetic potentials, functionally graded index, boundary constraint and size-dependent parameter on nonlinear dynamic response of the FG-MEE nanobeam. Simulation results have demonstrated the significance of the MEE coupling effect, which can be used as a guide to design smart and advanced devices for aerospace and industrial applications.

Static Response of Nanocomposite Electromagnetic Sandwich Plates with Honeycomb Core via a Quasi 3-D Plate Theory

This article investigates the static analysis of functionally graded electromagnetic nanocomposite sandwich plates reinforced with graphene platelets (GPLs) under hygrothermal loads. The upper and lower layers of nanocomposite face sheets are made of piezoelectromagnetic material with randomly oriented and uniformly disseminated or functionally graded (FG) GPLs throughout the thickness of the layers, while the core layer is made of honeycomb structures. The effective Young’s modulus of the face sheets of the sandwich plate is derived with the aid of the Halpin–Tsai model. While the rule of mixtures is incorporated to compute Poisson’s ratio and electric-magnetic characteristics of the sandwich plate’s upper and lower layers. The governing equations are obtained by a refined quasi-3-D plate theory, with regard to the shear deformation as well as the thickness stretching effect, together with the principle of virtual work. Impacts of the various parameters on the displacements and stresses such as temperature, moisture, GPLs weight fraction, external electric voltage, external magnetic potential, core thickness, geometric shape parameters of plates, and GPLs distribution patterns are all illustrated in detail. From the parameterized studies, it is significant to recognize that the existence of the honeycomb core causes the plate to be more resistant to the thermal condition and the external electric voltage because of the weak electricity and thermal conductivity of the honeycomb cells. Consequently, the central deflection decreases with increasing the thickness of the honeycomb core. Moreover, with varying the external electric and magnetic potentials, the deflection behavior of the sandwich structures can be managed; raising the electric and magnetic parameters contribute to an increment and decrement in the deflection, respectively.

Isogeometric Buckling Analysis of The Magneto-electro-elastic Foam Plates Resting on An Elastic Foundation

This study combines the isogeometric approach (IGA) and refined plate theory (RPT) with two variables to investigate buckling behavior of magneto-electro-elastic (MEE) foam plates resting on an elastic foundation. The pores in the MEE foam plates are arranged in three patterns: uniform, symmetric, and asymmetric distributions across the plate thickness. The elastic foundation supported by Winkler and Pasternak is utilized to approach computational model. The governing equations are derived by using RPT and Hamilton’s principle. The Non-Uniform Rational B-Splines (NURBS) basic functions in the IGA method are used to approximate the displacement fields, magnetic and electric potentials. The critical buckling load of the MEE foam plates is determined by solving the above governing equations with the help of the IGA. The study investigates and discusses the influence of various parameters such as the porosity distributions, porous coefficient, external electric voltage and magnetic potential, spring/shear coefficients of the elastic foundation, and the geometry of the MEE foam plates on the critical buckling load. The results show that these parameters significantly influence the buckling behavior of the MEE foam plates. This study provides valuable insights into the buckling behavior of magneto-electro-elastic foam plates and can inform the design of novel materials and structures with tailored properties.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.

Extensibility of External Magnetic Potential at High Latitudes - Antarctica

We investigated the external magnetic potential due to solar forcing, with nine years of data during 2001-2009, coveringthe deep solar minimum (2006-2009), at two stations: one is in the polar cap -Vostok (78º27'S, 106º52’E; mag. lat 83oS) andanother is in the subauroral region - Maitri (70º45'S, 11º43'E: mag. lat 67oS) in Antarctica. The significance of the work isassociated with space weather prediction and its impact on planet Earth. We used Advance Composition Explorer (ACE)satellite data for the aforesaid period for a thorough understanding of influences due to solar wind origin and to compare theparameter observed in these regions. We used the spherical cap harmonic analysis (SCHA) function as a tool. The inferenceindicates that at Vostok the magnitude is enhanced throughout and depicts a broad ambient external magnetic potential. Itseems to be essentially the intensification of the region 1 currents whereas at Maitri intense electric fields are producedduring geomagnetic perturbations which drive a system of disturbed time Region 2 currents over the quiet time currents.During this scenario in Maitri there are noticeable peaks or enhancements in the magnetic potential that can be observedmainly during geomagnetic disturbances. Hence the regression relation developed for external magnetic potentialcalculation, in terms of solar wind parameters agrees well with polar cap region and the area is relatively less exploredearlier, the present investigation can be expected to add knowledge about that regime.

Size-dependent nonlinear bending analysis of nonlocal magneto-electro-elastic laminated nanobeams resting on elastic foundation

Employing Reddy’s third-order shear deformation theory (RTSDT) and nonlocal elasticity theory, a nonlinear bending model of the nonlocal three-layer magneto-electro-elastic (MEE) laminated nanobeam resting on elastic foundation is established by considering von Karman’s geometrically nonlinear equations. Nonlinear higher order partial differential governing equations of MEE laminated nanobeams can be obtained employing Hamilton variational principle. The three-layered MEE laminated nanobeam is considered to have simply supported boundary condition in the present paper. Meanwhile, the electric and magnetic potential distributions in the laminated nanobeam are determined through Maxwell’s magnetic-electro equations and boundary conditions. The governing equations of laminated nanobeams are re-expressed in the dimensionless form by introducing the non-dimensional terms. Employing Galerkin method, the nonlinear higher order partial differential governing equations are simplified into lower order equations. Several cases are explored to indicate the effects of foundation parameters, nonlocal parameter, stacking sequence, external electric voltage and external magnetic potential on bending behaviors of MEE laminated nanobeams.

Wave Dispersion Analysis of Functionally Graded GPLs-Reinforced Sandwich Piezoelectromagnetic Plates with a Honeycomb Core

This paper studies wave propagation in a new structure composed of three layers. The upper and lower layers are made of a piezoelectromagnetic material reinforced with graphene platelets (GPLs) that may be uniformly disseminated or continuously varied throughout the thickness of the layers. To produce a lighter plate, the core layer is assumed to comprise honeycomb structures. The smart nanocomposite plate is exposed to external electric and magnetic potentials. The effective elastic modulus of the face layers of the sandwich plate is evaluated based on Halpin-Tsai model. Whereas, the mixture rule is utilized to calculate mass density, Poisson’s ratio and electric and magnetic properties of both upper and lower layers of the sandwich plate. The governing motion equations of the lightweight sandwich plate are obtained by refined higher-order shear deformation plate theory and Hamilton’s principle. These equations are solved analytically to obtain wave dispersion relations. Impacts of the geometry of plates, GPLs weight fraction, GPLs distribution patterns, piezoelectric properties, external electric voltage and external magnetic potential on the wave frequency and phase velocity of the GPLs lightweight plates are discussed in detail.

A new model to study magnetic-electric fields effects on bending of nano-scale magneto-electro-elastic beams

In this paper, a new 2D formulation is developed for analyzing the effects of magnetic and electric fields on the static bending behavior of magneto-electro-elastic (MEE) nanobeams subjected to transverse loading conditions. The nonlocal elasticity theory is utilized to consider the small-scale effect on the behavior of nanobeams. In order to increase the accuracy of the model to obtain more realistic predictions, a general displacement field is considered, which take into account a general form for the distribution of normal and shear strains in the nanobeam and to consider a general form for the distribution of magnetic and electric fields in the MEE nanobeam. Using the principle of minimum total potential energy, the governing equations of the MEE nanobeam under external magnetic, electric, and mechanical loadings are determined. An analytical solution is developed to investigate the effects of magnetic and electric fields on the bending of the MEE nanobeam. According to the numerical results, it is shown that the present method can predict the deflection of magneto-electro-elastic nanobeams more accurately in comparison with predictions of the shear deformation theories available in the literature, due to considering all in-plane and out of plane strain components and considering a general (not predefined) form for the distribution of electric and magnetic field through the thickness of MEE nanobeam. A detailed numerical study is presented to examine the influence of the nonlocal parameter, magnetic potential, electric potential, transverse and axial loadings, and aspect ratio on the deflection of magneto-electro-elastic nanobeam. The results revealed for BaTiO3–CoFe2O4 nanobeam that the maximum deflection of MEE nanobeam increases by applying a negative external magnetic potential as well as a positive external electric potential. Due to the excellent accuracy of the presented formulation, it can be used in the analysis and design of MEE nanostructures and as a benchmark for the investigation of the accuracy of other numerical models.

Thermal vibration and buckling of magneto-electro-elastic functionally graded porous nanoplates using nonlocal strain gradient elasticity

This study models and examines the thermal vibration and buckling behaviours of a porous nanoplate made of functional grading of barium-titanate and cobalt-ferrite. The porosity in the nanoplate is modelled with uniform and symmetrical porosity distribution functions. In the constitutive equation of the nanoplate, the strains are assumed to originate from classical mechanics, thermal expansion, electroelastic and magnetostrictive properties, nonlocal elasticity, and strain gradient elasticity. The motion equations of magneto-electro-elastic (MEE) nanoplate is obtained by Hamilton's principle. The study investigated the effects of thermal stresses, magneto-electro-elastic coupling, externally applied electric and magnetic field potential, nonlocal properties (nonlocal and material size parameters), and porosity volume fraction and function of porosity variation across thickness in free vibration and buckling behaviour of the nanoplate. According to the analysis results, the dimensionless frequencies decrease as the barium-titanate ratio in the nanoplate increases, and the frequencies increase as the cobalt-ferrite ratio increases, depending on the material grading index. While temperature increased and porosity ratio decreased dimensionless frequencies, external magnetic potential increased dimensionless frequencies. On the other hand, the application of electric potential causes a slight increase in dimensionless frequencies compared to the effect of magnetic potential.

Size-dependent free vibration and buckling analysis of magneto-electro-thermo-elastic nanoplates based on the third-order shear deformable nonlocal plate model

In this paper, a size-dependent nanoplate model is developed to describe the free vibration and buckling behaviors of magneto-electro-thermo-elastic (METE) rectangular nanoplates. It is assumed that the METE nanoplate is subjected to external electric voltage, external magnetic potential, and uniform temperature rise. The nonlocal elasticity theory along with the third-order shear deformation plate theory is employed for the size-dependent mathematical modeling of nanoplates. The presented model has two advantages over available models: (1) the need for the correction factor is bypassed and (2) it can be successfully applied to thick nanoplates. The governing equations and the corresponding boundary conditions are derived using the Hamilton’s principle which are then discretized on the space domain based on the generalized differential quadrature (GDQ) method. Afterward, an efficient numerical Galerkin procedure is adopted to reduce the discretized equations into Duffing-type ordinary differential equations. Numerical results are presented to examine the influences of nonlocal parameter, length-to-thickness ratio, temperature rise, external electric potential, external magnetic potential and type of boundary condition on the free vibration, and buckling behaviors of METE nanoplates. It is revealed that frequency and critical buckling load of nanoplates are dependent on magneto-electro-mechanical loadings, whereas they are less dependent on thermal loading.

Analysis of Electromagnetic Effects on Vibration of Functionally Graded GPLs Reinforced Piezoelectromagnetic Plates on an Elastic Substrate

A new nanocomposite piezoelectromagnetic plate model is developed for studying free vibration based on a refined shear deformation theory (RDPT). The present model is composed of piezoelectromagnetic material reinforced with functionally graded graphene platelets (FG-GPLs). The nanocomposite panel rests on Winkler–Pasternak foundation and is subjected to external electric and magnetic potentials. It is assumed that the electric and magnetic properties of the GPLs are proportional to those of the electromagnetic materials. The effective material properties of the plate are estimated based on the modified Halpin–Tsai model. A refined graded rule is introduced to govern the variation in the volume fraction of graphene through the thickness of the plate. The basic partial differential equations are provided based on Hamilton’s principle and then solved analytically to obtain the eigenfrequency for different boundary conditions. To check the accuracy of the present formulations, the depicted results are compared with the published ones. Moreover, impacts of the variation in elastic foundation stiffness, plate geometry, electric potential, magnetic potential, boundary conditions and GPLs weight fraction on the vibration of the smart plate are detailed and discussed.

Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus

<p style='text-indent:20px;'>Although the nuclear fusion process has received a great deal of attention in recent years, the amount of mathematical analysis that supports the stability of the system seems to be relatively insufficient. This paper deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the <i>Vlasov-Maxwell</i> system in a two-dimensional annulus when a huge (<i>but finite-in-time</i>) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The authors hope that this work is a step towards a more generalized work on the three-dimensional Tokamak structure. The highlight of this work is the physical assumptions on the external magnetic potential well which remains finite <i>within a finite time interval</i> and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the <i>Vlasov-Maxwell</i> system.</p>

Tunable Bandgaps in Phononic Crystal Microbeams Based on Microstructure, Piezo and Temperature Effects

A new model of non-classical phononic crystal (PC) microbeam for the elastic wave bandgap generation is provided, incorporating microstructure, piezomagnetism, piezoelectricity and temperature effects. The wave equation of a general magneto–electro–elastic (MEE) phononic crystal microbeam is derived, which recovers piezoelectric- and piezomagnetic-based counterparts as special cases. The piezomagnetic and piezoelectric materials are periodically combined to construct the PC microbeam and corresponding bandgaps are obtained by using the plane wave expansion (PWE) method. The effects of the piezomagnetism, piezoelectricity, microstructure, geometrical parameters and applied multi-fields (e.g., external electric potential, external magnetic potential, temperature change) on the bandgaps are discussed. The numerical results reveal that the bandgap frequency is raised with the presence of piezo and microstructure effects. In addition, the geometry parameters play an important role on the bandgap. Furthermore, large bandgaps can be realized by adjusting the external electric and magnetic potentials at micron scale, and lower bandgap frequency can be realized through the temperature rise at all length scales.

Uniform Error Bounds of Time-Splitting Methods for the Nonlinear Dirac Equation in the Nonrelativistic Regime without Magnetic Potential

Super-resolution of the Lie-Trotter splitting ($S_1$) and Strang splitting ($S_2$) is rigorously analyzed for the nonlinear Dirac equation without external magnetic potentials in the nonrelativistic regime with a small parameter $0<\varepsilon\leq 1$ inversely proportional to the speed of light. In this regime, the solution highly oscillates in time with wavelength at $O(\varepsilon^2)$. The splitting methods surprisingly show super-resolution, i.e. the methods can capture the solution accurately even if the time step size $\tau$ is much larger than the sampled wavelength at $O(\varepsilon^2)$. Similar to the linear case, $S_1$ and $S_2$ both exhibit $1/2$ order convergence uniformly with respect to $\varepsilon$. Moreover, if $\tau$ is non-resonant, i.e. $\tau$ is away from certain region determined by $\varepsilon$, $S_1$ would yield an improved uniform first order $O(\tau)$ error bound, while $S_2$ would give improved uniform $3/2$ order convergence. Numerical results are reported to confirm these rigorous results. Furthermore, we note that super-resolution is still valid for higher order splitting methods.

Deconfinement of heavy quarks at finite density and strong magnetic field

We study the simultaneous effect of the external magnetic field and finite chemical potential on deconfinement of heavy quarks, probed by the Polyakov loop and its fluctuations. We calculate the Polyakov loop, its real and imaginary susceptibilities, their ratio, as well as the heavy quark and anti-quark entropies. We find that these quantities are sensitive probes of deconfinement. Especially, the real Polyakov loop susceptibility and heavy quark and anti-quark entropies diverge at the critical point.