Expression For Potential Energy Research Articles

Analysis and experiments of bi-stable laminate configuration control based on MFC piezoelectric actuation

This paper studies the configuration change control and the snap-through phenomenon of bi-stable laminate with the MFC piezoelectric plate. The kinetic and potential energy expressions of the piezoelectric bi-stable laminate were obtained based on the von-Karman hypothesis. The constitutive equations of the system were given using a six-parameter displacement model, and the potential function of the system was derived using Hamilton’s principle. The configuration changes of the piezoelectric bi-stable laminate under different geometrical conditions were analyzed. The influence of voltage on the bi-stable laminate configuration was studied by applying different driving voltages. The snap-through energy variation of the piezoelectric bi-stable laminate under different geometrical conditions was investigated. In the experimental part, the changes in bi-stable laminate configuration with different geometrical conditions and after attaching the piezoelectric plate were compared. Different driving voltages were applied to measure the difference in piezoelectric bi-stable laminate configuration, and the critical snap-through voltages were recorded. The measured changes in bi-stable laminate configuration and snap-through voltage at different voltages prove the correctness of the theoretical presentation.

A New Analytical Approach for Nonlinear Global Buckling of Axially Compressed and Tensiled Sandwich Toroidal Shell Segments with CNTRC Coatings and Corrugated Core in Thermal Environment

In this paper, the nonlinear global buckling and postbuckling responses of axially compressed and tensiled sandwich toroidal shell segments with functionally graded carbon nanotube-reinforced composite (FG-CNTRC) coatings and corrugated core in the thermal environment are investigated. Three distributed types of FG-CNTRC coatings with two geometrical types of corrugated core, and three shell types including convex, concave, and cylindrical shells are considered. By adding the thermal forces to the governing equations, a homogenization model for corrugated structures is developed to simulate the thermal and mechanical behavior of the trapezoidal and round corrugated core. The potential energy expression of sandwich shells is established according to the Donnell shell theory with the nonlinearities of deflection, an algorithm to solve the governing equation is developed using the Ritz energy method, and three equilibrium equations are obtained respecting three amplitudes of deflection. The axial critical buckling loads and postbuckling curves of shells can be achieved after some calculations. The numerical examples prove the beneficial effects of corrugated core and CNTRC coatings on nonlinear global buckling responses of sandwich toroidal shell segments significantly.

Interaction potential between coplanar uniformly charged disk and ring

We consider a system made up of a uniformly charged disk and a uniformly charged ring that are coplanar with each other. The disk and ring have arbitrary radii and contain arbitrary amounts of net charge that is spread uniformly over area and length, respectively. This study is seen as the first necessary step towards a classical model treatment of a two-dimensional system of electrons in a weak perpendicular magnetic field. In this scenario, the uniformly charged disk represents the neutralizing background in a jellium approximation while a uniformly charged ring is viewed as a classical approximation to the cyclotron motion of the electron in presence of a uniform perpendicular magnetic field. In this work, we obtain an exact integral expression for the interaction potential energy of the system as a function of the separation distance between the centers of the two bodies and discuss various emerging features of this model.

On the Modernisation of Weber’s Electrodynamics

This work is an attempt to modernise Weber’s electrodynamics to make it compatible with the high-velocity regime, and with the existence of a limiting velocity, c. For this purpose, starting from the law of energy conservation and the mass–energy equivalence, new expressions for potential energy and for kinetic energy are derived jointly which are consistent with an ultimate velocity of the value of c. The new potential energy, already reported by Phipps, becomes Weber’s expression in the limit of low velocities. The new kinetic energy differs from the relativistic expression, but, like the latter, it also becomes the Newtonian expression in the limit of low velocities. New expressions for force and linear momentum are also derived which complete a new mechanics. Phipps’ potential energy and new kinetic energy are applied to the problem of two interacting charges in a radial motion and orbital motion. The new framework is also applied to the problem of a charge moving between the two plates of a charged capacitor, obtaining a result similar to that obtained by means of Maxwell–Lorentz electromagnetism and relativistic mechanics. The metaphysical considerations that clearly differentiate the conventional framework from the new framework proposed here are discussed.

A Novel Analytical Approach for Nonlinear Thermo-Mechanical Buckling of Higher-Order Shear Deformable Porous Circular Plates and Spherical Caps with FGM Face Sheets

A novel analytical approach for nonlinear thermo-mechanical buckling of higher-order shear deformable porous circular plates and spherical caps with functionally graded material (FGM) face sheets resting on Pasternak elastic foundation is presented in this paper. The circular plates and spherical caps are assumed to be subjected to uniformly distributed external pressure and/or uniformly distributed thermal loads, and the nonlinear higher-order shear deformation theory (HSDT) is used for largely thick plates and caps with the shell-foundation interaction modeled by Pasternak elastic foundation. The caps are assumed to be shallow with clamped boundary conditions. The total potential energy expression of structures is established and the Ritz energy method is used to solve the problem directly from the total potential energy expression. The expressions between external pressure–deflection, thermal load–deflection, and thermo-mechanical combined load–defection can be obtained using the iterative algorithms. The critical buckling loads and postbuckling behavior of plates/caps are investigated numerically. Significant effects of foundation, porosity, structure parameters on the nonlinear thermo-mechanical responses of circular plates and spherical caps are numerically investigated and discussed, and the complex tendencies of postbuckling strength of plates and caps are obtained.

Stress Analysis of Transversely Loaded Isotropic Three-Dimensional Plates Using a Polynomial Shear Deformation Theory

An exact solution for the bending attributes of a thick rectangular plate under transverse loading is modeled herein using three-dimensional (3-D) elasticity plate theory and fourth-order polynomial shear deformation function. Precluding coefficients of shear correction, this model captured the effect of shear deformation along with the transverse normal strain stress. The expression for total potential energy was derived from a 3-D kinematic and constitutive relation the equilibrium equation was developed and employed from the energy functional transformation to get the relationship between slope and deflection. Exact polynomial functions were obtained from the outcome of the equilibrium equation and with the aid of the direct variation approach, the coefficient of deflection of the plate was generated from the governing equation. The expression for computing the displacement, bending moments, and stress components along the three axes of the plate was established from these solutions for the assessment of the bending characteristics of a rectangular plate. The result of a simply supported at one edge, free at one edge and clamped at the two outer edges (SCFC) was evaluated using the obtained functions in this study. The report of this study confirms the exactness and consistency of the 3-D model unlike the refined plate theories applied by previous authors in the available literature. The value of 8.05% is the comprehensive average percentage variation of the values for center deflection obtained by Onyeka and Okeke (2020) and Gwarah (2019). It is established that at the 92 % confidence level, this model is worthy of adoption for safe, cost-effective and accurate bending analysis of thick plates of any support condition.

Potential energy surface, quartic force fields and vibrational levels of chlorine dioxide anion

Potential energy surface was calculated for the ground electronic state of the triatomic ion ClO2− using the coupled-cluster method CCSD(T). Calculations were carried out for 1200 points on the potential energy surface, and the calculated points were fitted to the potential energy expression in terms of the internal coordinates, from which the quadratic, cubic and quartic force fields were determined. Using the second-order rovibrational perturbation theory, harmonic vibrational frequencies, anharmonicity constants and several other spectroscopic parameters were calculated, and accurate fundamental vibrational frequencies were obtained. Also, the energies of 30 lowest vibrational levels were calculated using the anharmonicity constants.

Modelling the impact of argon atoms on a tungsten surface

Sputtering from plasma-facing surfaces upon particle impact is an important process in material science. It is especially relevant in the diverter region of fusion devices, which nearly always consist of tungsten. Besides the main plasma components, argon is used in fusion devices to improve energy confinement. As a consequence, hot Ar atoms interact with W surfaces and can cause sputtering and other material degrading events. Atomistic simulations of the plasma-wall interactions make it possible to carry out a detailed analysis of sputtering, reflection, and retention processes. We report the results of molecular dynamics simulations with neural network potential energy expressions modelling the bombardment of tungsten samples by argon atoms in the energy range from 100 to 800 eV. The obtained sputtering results are in good agreement with available literature data. Furthermore, our data provide additional insight into atomic details of the processes involved in sputtering. We also investigate the effect of surface temperature on sputtering and reflection probabilities, which significantly affects the irradiation process at higher impact energies.Graphical abstract

Application of variation method in three dimensional stability analysis of rectangular plate using various exact shape functions

In this paper, a polynomial and trigonometric shape function are developed for the three-dimensional (3-D) stability analysis of a thick rectangular plate. This study has evaluated the effect of aspect ratio of the critical buckling load of a plate that is clamped on the first edge, free at the third edge, with the second and fourth edges simply supported respectively (CSFS) using a variational method. An expression of potential energy of thick plate was formulated using 3-D elastic principles thereafter, a compatibility equation of 3-D plate was derived through energy equation transformation to get the relations between the rotations and deflection. The solution of compatibility equations yields the exact plates shape function which is derived in terms of trigonometric and polynomial displacement and rotations. Similarly, by minimizing the energy equation with respect to the deflection, the direct governing equation was formulated. The solution of governing equation yields the deflection coefficient of the plate. By minimizing the potential energy equation with respect to deflection coefficient after the action deflection and rotations equation were substituted into it, a more realistic formula for calculation of the critical buckling load is established. The proposed method unlike the refined plate theory (RPT), considered all the six stress elements in the analysis. The result showed that the critical buckling loads from the present study using polynomial are slightly higher than those obtained using trigonometric theories signifying the more exactness of the latter. The result of the present study using the established 3-D model for both functions is satisfactory and closer to exact solution compared to the two- dimensional (2-D) RPT. The overall average percentage differences between the two functions recorded are 6.4%. This shows that at about 94% both approaches are the same and can be applied with confidence in the stability analysis of any type of plate with the boundary condition.

An anisotropic negative thermal expansion metamaterial with sign-toggling and sign-programmable Poisson’s ratio

AbstractObjectivesA mechanical metamaterial is introduced herein by drawing inspiration from an Aztec geometric pattern. This metamaterial deformation mechanism for Poisson’s ratio and Young’s modulus is based on non-rotating rhombi with rotating triangles, while the shear modulus analysis herein is based on rotating rhombi with non-rotating triangles, hence ‘partially rotating rigid units’.MethodsThe coefficient of thermal expansion was obtained by equating the potential energy expressions from the simple harmonic motion and from the principle of energy equipartition, while the effective-moduli were acquired by equating the strain energy from rotational stiffness with that from the strain energy of deformation based on an assumed homogenized continuum. Due to the zero and extreme Poisson’s ratio based on infinitesimal deformation, the finite approach was employed.ResultsResults indicate that the proposed metamaterial exhibits anisotropic negative thermal expansion with sign-switching Poisson’s ratio when applied stress along one axis is reversed. The Poisson’s ratio for loading in another axis is undefined under tension but can be programmed to exhibit either sign when compressed. The Young’s and shear moduli are directly governed by the rotational stiffness and strongly influenced by the extent of rotation, followed by the aspect ratio of the rotating units.ConclusionDue to its uniqueness, the currently considered mechanical metamaterial can be used under specific requirements which are difficult to be attained by other materials with negative properties.

Physical Application of Taylor Formula

Taylor's formula is an important content in advanced mathematics, and it is also an indispensable mathematical tool in the study of function limits and error estimation. In this paper, Taylor's formula is used to deduce the electric potential energy of the charge system in the applied electric field and to prove the Noether's theorem of the point particle system under the action of conservative force. Finally, the expression of potential energy of multi electrode in external electric field and the reason why Noether's theorem is established in conservative system are given, and the superposition principle of potential energy of point particle system is indirectly explained.

A unified strong spectral Tchebychev solution for predicting the free vibration characteristics of cylindrical shells with stepped-thickness and internal–external stiffeners

In this paper, a strong spectral Tchebychev (ST) solution is developed to investigate the free vibration behaviors of the cylindrical shell with stepped-thickness (CS-S) and cylindrical shell with internal–external stiffeners (CS-IES) subject to arbitrary boundary constraints. The spring parameter technique is used to impose combined connections and arbitrary boundaries. The potential and kinetic energy expressions of CS-S and CS-IES are derived based on the framework of the first-order shear deformation theory (FSDT). Finally, the unified governing differential equations of CS-S and CS-IES are acquired by the Hamilton’s variational principle. According to the Gauss–Lobatto point discretization process, the Tchebyshev polynomials are used to perform spectral expansion on the axial variables of admissible displacement functions. Circumferential variables are represented by sine and cosine series. In this way, the unified free vibration mathematical model of CS-S and CS-IES is obtained. On the basis of confirming that the model results have reached the state of convergence, the numerical results are compared with those calculated by the reference and the finite element method, which fully validates that the strong spectral Tchebychev model obtained in this paper is feasible for analyzing the free vibration characteristics of CS-S and CS-IES subject to arbitrary boundary constraints. Last but not least, the paper carries out the parametric analysis of how the change of the geometric parameters of the stiffeners will affect the natural frequencies of CS-IES.

Derivation of general relativistic gravitational potential energy using principle of equivalence and gravitational time dilation

Einstein’s theory of general relativity, which has been experimentally proved to be a true theory of gravity does not need gravitational potential energy to predict the trajectory of particles in space. This is because general relativity is a purely geometric theory. Objects move along the geodesics in the curved space–time. The energy–momentum tensor that warps space–time as per Einstein’s field equations takes into account only the energy–momentum of matter and radiation. Thus, gravitational potential energy does not come into the picture in Einstein’s theory of gravity and its role is taken over by the curvature of space–time. However, a general relativistically correct expression for gravitational potential energy is required for energy conservation and some energy-based approaches in physics. Conventionally, the correct form of gravitational potential energy is derived by using the full mathematical formality of general relativity. In this paper, we derive the same general relativistic expression for gravitational potential energy simply by using the principle of equivalence and gravitational time dilation.

A model liquid crystal cell in an RC circuit

A nematic liquid-crystal cell subject to an electric field created by electrodes held at constant potential is modeled as a variable capacitor in an RC circuit. The state of the system is characterized in terms of the director field in the cell and the charge on the electrodes. A dynamical system is developed that couples director dynamics in the cell (with no fluid flow) and charge dynamics in the circuit. The dynamical equations are derived from expressions for the total potential energy of the system and a dissipation involving a single rotational viscosity for the director plus Joule heating associated with current in the circuit. An effort is made to quantify effects, in particular the widely differing time scales for the processes involved, and numerical illustrations are given. The exercise illuminates aspects of the modeling of equilibrium states of such a system.

Mechanical properties of group IV single-walled nanotubes: a finite element approach based on the density functional theory.

In this article, the density functional theory is applied to characterize the mechanical properties of single-walled nanotubes of group IV of the periodic table. These materials include carbon nanotube, silicon nanotube, germanium nanotube, and stanene nanotube. (10,10) armchair nanotube is selected for the investigation. By establishing a link between potential energy expressions in molecular and structural mechanics, a finite element approach is proposed for modeling nanotubes. In the proposed model, the nanotubes are considered as an assemblage of beam elements. Young's modulus of the nanotubes is computed by the proposed finite element model. Young's modulus of carbon, silicon, germanium, and tin nanotubes are obtained as 1029, 159.82, 83.23, and 83.15GPa, respectively, using the density functional theory. Also, the finite element approach gives the values as 1090, 154.67, 85.2, and 82.6GPa, respectively. It is shown that the finite element model can predict the results of the density functional theory with good accuracy.

VIBRATION CHARACTERISTICS OF NONUNIFORM BLADES MADE OF FUNCTIONALLY GRADED MATERIAL

The purpose of this study is to examine the vibration characteristics of a rotating blade whose material distribution varies in the spanwise direction. Formulations for functionally graded materials and beam structural models are carried out in detail and the results are displayed in several figures and tables which is a significant source of information for the authors working in this area. Different parameters such as angular speed, radius of the hub, material properties, power law index parameter, boundary conditions and slenderness ratio are considered in the formulation. Finite Element Method where the element matrices are obtained from potential and kinetic energy expressions is applied as the solution procedure. The calculated results are demonstrated in various tables and figures where it is observed that there is a good agreement with literature.

Um modelo matemático para a interação de um vírus com a membrana plasmática da célula

O entendimento das interações dos vírus com as membranas celulares é de fundamental importância para a proposta de vacinas e tratamentos de doenças causadas por esse tipo de contaminação. A exemplo disso, cita-se o caso da pandemia causada pelo novo coronavírus causador da Covid-19, que no ano de 2020 colocou um terço do mundo em situação de quarentena, causando milhares de mortes e prejuízos econômicos em todo o planeta. Nesse contexto, o presente estudo propõe-se a construir um modelo matemático para a interação de um vírus com a membrana plasmática da célula de um mamífero, que conduz a um problema matemático com condições de contorno. Fazendo uso das funções de Green, o modelo foi capaz de responder qual é a expressão geral para energia potencial eletrostática desta interação em termos das funções de Bessel e os coeficientes de Wigner. Pode-se especular, ancorado nos resultados apresentados pelo modelo, que um vírus tem que apresentar uma carga efetiva resultante em solução com pH da ordem de 7 para se aproximar da membrana celular, enquanto um vírus eletricamente neutro não deve apresentar a capacidade de interagir e adentrar em membranas. Os valores para a energia de interação e para a força entre a membrana e o aglomerado oferecem a ordem de grandeza das distâncias em que tal interação é efetiva.

A penalty function based delamination model for postbuckling analysis of composite plates with delamination

A novel technique for modeling the mechanical behavior of partially delaminated composite plates is proposed and adopted for addressing the postbuckling problem of such plates under compressive in-plane loading. The new method considers a laminate as two sublaminates on the two sides of the plane of an existing delamination. The model assumes that the sublaminates are bonded to each other except in the delaminated region. The sublaminates are separately modeled using a first-order-shear deformation theory and to address both pre and postbuckling regimes, a strain definition capable of accommodating large rotations is used. The equilibrium equations are obtained using a modified Ritz method in conjunction with the stationary total potential energy principle. The modified Ritz method makes use of a combination of continuous and noncontiguous functions to accurately represent highly local deformations that can be expected in a delaminated region. The bonding between the sublaminates is simulated using a penalty method and by adding the energy terms corresponding to an elastic adhesion between the sublaminates to the expression of the total potential energy. Numerical examples are presented and compared with the results obtained using 3D finite element analyses.

Anisotropic Keldysh interaction

We generalize the classic calculations by Rytova and Keldysh of screened Coulomb interaction in semiconductor thin films to systems with anisotropic permittivity tensor. Explicit asymptotic expressions for electrostatic potential energy of interaction in the weakly anisotropic case are found in closed analytical form. The case of strong in-plane anisotropy, however, requires evaluation of the inverse Fourier transform of 1/(k+Akx2+Bky2), which, at present, remains unresolved.

A New Approach to the Potential Energy of Solids

A general theory is developed to investigate the expression for potential energy without using (i) empirical results and (ii) process of parameterizing. The simple approach of thermodynamics is adopted to obtain the expressions for the interaction energy of solids in terms of interatomic separation and crystal volume. The new findings have been applied to obtain the expressions for bulk modulus and pressure. The variation of the potential energy function, which provides a means to understand the stability of a crystal has been found in excellent agreements to the earlier results. The use of obtained harmonic and anharmonic force constants may be of much help to understand the dynamical behavior of solids with special reference to the many-body theories.