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1

A simplified elliptical function solution for coupled nonlinear vibration of multilayer graphene sheets

Venkatraman G., Suji D.

Journal of Theoretical and Applied Mechanics

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2022

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Vol. 60 nr 1

3--16

EN

This paper presents a simple exact solution for coupled nonlinear free vibration analysis of Multilayer Graphene Sheets (MLGS) using elliptical functions. Though elliptical functions are used in nonlinear dynamics, they are employed to find the exact solution of a coupled system for the first time. The nonlinear dynamic equation including geometric nonlinearity and Eringen nonlocal theory is uncoupled by elliptical functions, and exact solutions for simply supported boundary conditions are obtained. The results are compared with the harmonic balance method. The nonlinear frequency of MLGS is studied for its effects with a small scale parameter, and the linear and nonlinear van der Waals force.

2

Thermal analysis of longitudinal a porous fin with temperature-dependent internal heat generation using the variation of parameters method

Güngör Osman, Arslantürk Cihat

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 3

5--16

EN

The present study investigates the thermal performance of longitudinal a porous fin with temperature-dependent internal heat generation. The Darcy model is utilized to obtain the differential form of the governing equation that solves the nonlinear temperature distribution equation using the method of variation of parameters. Although this method is applied to solve both linear and nonlinear differential equations, there exist rare applications of this method to solve nonlinear heat transfer problems. In the present study, we applied the method to estimate the thermal analysis of the porous fin exposed to convection. The heat generation is assumed as a function of temperature. The effects of the convection parameter Nc, internal heat generation ɛ, porosity Sh, and generation number G parameter on the dimensionless temperature distribution are discussed in detail. The accuracy of the variation of parameters method is verified through comparison with homotopy perturbation method and the Matlab bvp4c solver (NUM). The results have disclosed that the variation of parameters method can be used as a very effective and practical approach for further studies of the porous medium.

3

Numerical and analytical calculations of the free surface flow between two semi-infinite straights

Chedala Fatma Zohra, Amara Abdelkader, Meflah Mabrouk

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 4

21--32

EN

The problem of two-dimensional flow with the free surface of the jet in a region between two semi-infinite straights intersections at point O is calculated analytically for each angle Beta and numerically for each of the various values of the Weber number and angle Beta. By assuming that the flow is potential, irrotational and that the fluid is incompressible and inviscid, and by taking account only the surface tension for a numerical method using the series truncation, and without the effect of gravity and surface tension for the analytic method utilize the hodograph transformation. The obtained results confirmed a good agreement between them when the Weber number tends to infinity, and the comparison of these surface shapes is illustrated.

4

Group invariant solution of some time fractional evolution equations

Mandal H., Bira B.

Journal of Applied Mathematics and Computational Mechanics

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2018

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Vol. 17, nr 4

35--46

EN

In this paper, we consider some classes of a system of nonlinear fractional differential equations (FDEs) arising in some important physical phenomena. Using symmetry group of transformations, the given systems of fractional partial differential equations (FPDEs) are reduced to systems of fractional ordinary differential equations (FODEs). Further, using the group invariant condition, we solve the reduced systems of FODEs and exact solutions of the given equations are constructed. Finally, the physical significance of the solutions are investigated graphically based on the exact solutions in order to highlight the importance of the study.

5

A Fundamental Study on the Free Vibration of Geometrical Nonlinear Cantlever Beam using an Exact Solution and Experimental Investigation

Jamal-Omidi M., Shayanmehr M., Sazesh S.

Archive of Mechanical Engineering

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2018

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Vol. LXV, nr 1

65--82

EN

Two fundamental challenges in investigation of nonlinear behavior of cantilever beam are the reliability of developed theory in facing with the reality and selecting the proper assumptions for solving the theory-provided equation. In this study, one of the most applicable theory and assumption for analyzing the nonlinear behavior of the cantilever beam is examined analytically and experimentally. The theory is concerned with the slender inextensible cantilever beam with large deformation nonlinearity, and the assumption is using the first-mode discretization in dealing with the partial differential equation provided by the theory. In the analytical study, firstly the equation of motion is derived based on the theory of large deformable inextensible beam. Then, the partial differential equation of motion is discretized using the Galerkin method via the assumption of the first mode. An exact solution to the obtained nonlinear ordinary differential equation is developed, because the available semi analytical and approximated methods, due to their limitations, are not always sufficiently reliable. Finally, an experiment set-up is developed to measure the nonlinear frequency of oscillations of an aluminum beam within a domain of initial displacement. The results show that the proposed analytical method has excellent convergence with experimental data.

6

Exact solution for large amplitude flexural vibration of nanobeams using nonlocal Euler-Bernoulli theory

Nazemnezhad R., Hosseini-Hashemi S.

Journal of Theoretical and Applied Mechanics

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2017

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Vol. 55 nr 2

649--658

EN

In this paper, nonlinear free vibration of nanobeams with various end conditions is studied using the nonlocal elasticity within the frame work of Euler-Bernoulli theory with von K´arm´an nonlinearity. The equation of motion is obtained and the exact solution is established using elliptic integrals. Two comparison studies are carried out to demonstrate accuracy and applicability of the elliptic integrals method for nonlocal nonlinear free vibration analysis of nanobeams. It is observed that the phase plane diagrams of nanobeams in the presence of the small scale effect are symmetric ellipses, and consideration the small scale effect decreases the area of the diagram.

7

Residual Power Series Method for Fractional Diffusion Equations

Kumar A., Kumar S., Yan S.-P.

Fundamenta Informaticae

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2017

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Vol. 151, nr 1/4

213--230

EN

In this article, improved residual power series method (RPSM) is effectively implemented to find the approximate analytical solution of a time fractional diffusion equations. The proposed method is an analytic technique based on the generalized Taylor’s series formula which construct an analytical solution in the form of a convergent series. In order to illustrate the advantages and the accuracy of the RPSM, we have applied the method to two different examples. In case of first example, different cases of initial conditions are considered. Finally, the solutions of the time fractional diffusion equations are investigate through graphical representation, which interpret simplicity, accuracy and practical usefulness of the present method.

8

Exact solution to the structural instability of a parallel array of mutually attracting identical simply-supported plates

Wang X., Liu L.

Archives of Mechanics

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2017

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Vol. 69, nr 3

243--256

EN

By means of the theory of elasticity, we investigate the structural instability of a parallel array of identical simply-supported plates. One plate interacts with the neighboring plates through surface attractive forces. The proposed method is based on the 2 × 2 transfer matrix for a plate and on the solution of the generalized eigenvalue problem for the plate array. Analytical expressions of the critical interaction coefficients for two, three and four interacting plates are obtained when the end-effect of the plates at the ends of the parallel array and the surface energy of the plates are ignored. The influence of the end-effect and surface energy on the critical interaction coefficient is also numerically studied. Our solution is valid whether the plates are thin or extremely thick.

9

Green’s function for a multifield material with a heat source

Rogowski B.

Journal of Theoretical and Applied Mechanics

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2016

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Vol. 54 nr 3

743--755

EN

Green’s functions for a multifield material subjected to a point heat source are presented in an explicit analytical form. The study concerns the steady-state thermal loading infi- nite region, half-space region and two-constituent magneto-electro-thermo-elastic material region. The new mono-harmonic potential functions, obtained by the author, are used in the analysis. The elastic displacement, electric potential, magnetic potential and induced by those coupled multifield physical quantities, caused by internal or external heat sources, are limited and presented in a very useful form, exactly and explicitly.

10

Exact solution of fin problem with linear temperature-dependent thermal conductivity

Kader A. H. A., Latif A. S. A., Nour H. M.

Journal of Applied Mathematics and Computational Mechanics

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2016

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Vol. 15, nr 4

51--61

EN

In this paper, we obtain the general exact solution of a nonlinear fin equation which governs heat transfer in a rectangular fin with linear temperature-dependent thermal conductivity using the partial Noether method. The relationship between the fin efficiency and the thermo-geometric fin parameter is obtained. Additionally, we obtained the relationship among the fin effectiveness, the thermo-geometric fin parameter and the Biot number.

11

Symmetry reductions and exact solutions to the Sharma–Tasso–Olever equation

Zhang Y.

Journal of Applied Analysis

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2016

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Vol. 22, nr 2

107--114

EN

In the present paper, the Sharma–Tasso–Olever (STO) equation is considered by the Lie symmetry analysis. All of the geometric vector fields to the STO equation are obtained, and then the symmetry reductions and exact solutions of the equation are investigated. Our results witness that symmetry analysis is a very efficient and powerful technique in finding the solutions of the proposed equation.

12

Hypothetical analysis for peristaltic transport of metallic nanoparticles in an inclined annulus with variable viscosity

Nadeem S., Sadaf H.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2016

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Vol. 64, nr 2

447--454

EN

The main objective of this article is to present a mathematical model for peristaltic transport in an inclined annulus. In this analysis, two-dimensional flow of a viscous nanofluid is observed in an inclined annulus with variable viscosity. Copper as nanoparticle with blood as its base fluid has been considered. The inner tube is unifom or rigid, while the outer tube takes a sinusoidal wave. Governing equations are solved under the well-known assumptions of low Reynolds number and long-wavelength. Exact solutions have been established for both velocity and nanoparticle temperature. The features of the peristaltic motion are explored by plotting graphs and discussed in detail.

13

Contact problem of conducting and heated punch on a multifield foundation

Rogowski B.

International Journal of Applied Mechanics and Engineering

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2015

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Vol. 20, no. 3

531--564

EN

The solution for a multifield material subjected to temperature loading in a circular region is presented in an explicit analytical form. The study concerns the steady – state thermal loading infinite region (heated embedded inclusion), half – space region and two – constituent magneto – electro – thermo – elastic material region. The new mono – harmonic potential functions, obtained by the author, are used in the analysis of punch problem. The more interested case in which the contact region is annular is analyzed. By using the methods of triple integraf equations and series solution technique the solution for an indentured multifield substrate over an annular contact region is given. The sensitivity analysis of obtained indentation parameters shows some interesting points. In particular, it shows that the increasing of the applied electric and magnetic potentials reduces the indentation depth in multifield materials.

14

A three-dimensional exact state-space solution for cylindrical bending of continuously non-homogenous piezoelectric laminated plates with arbitrary gradient composition

Lezgy-Nazargah M.

Archives of Mechanics

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2015

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Vol. 67, nr 1

25--51

EN

Most of exact solutions reported for the analysis of functionally graded piezoelectric (FGP) plates are based on the assumption, that the graded plate consists of a number of layers, where the material properties within each layer are invariant. The limited works that consider the continuous variation of electro-mechanical properties are restricted to FGP materials with the exponent-law dependence on the thickness-coordinate. In the present paper, a three-dimensional (3D) exact solution is presented for cylindrical bending of the FGP laminated plates based on the state space formalism. In contrast to the other reported solutions which are restricted to FGP materials with the exponent-law dependence on the thickness-coordinate, the present exact solution considers materials with arbitrary compositional gradient. Moreover, no assumption on displacement components and the electric potential along the thickness direction of FGP layers is introduced. Regardless of the number of layers, equations of motion, charge equation, and the boundary and interface conditions are satisfied exactly. The obtained exact solution can be used to assess the accuracy of different FGP laminated plate theories and/or for validating finite element codes.

15

Solitary wave solution to the nonlinear evolution equation in cascaded quadratic media beyond the slowly varying envelope approximations

Sarma A K, Biswas A.

Optica Applicata

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2014

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Vol. 44, nr 2

205--212

EN

We report an exact bright and dark soliton solution to the nonlinear evolution equation derived by MOSES and WISE (Phys. Rev. Lett. 97, 2006, 073903) for cascaded quadratic media beyond the slowly varying envelope approximations. The integrability aspects of the model are addressed. The travelling wave hypothesis as well as the ansatz method are employed to obtain an exact 1-soliton solution. Both bright and dark soliton solutions are obtained. The corresponding constraint conditions are obtained in order for the soliton solutions to exist.

16

Anti – plane crack emanating from the interface in a bounded smart PEMO- elastic structure

Rogowski B.

International Journal of Applied Mechanics and Engineering

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2013

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Vol. 18, no. 4

1165--1199

EN

The magnetoelectroelastic analysis of two bonded dissimilar piezo-electro-magneto-elastic ceramics with a crack perpendicular to and terminating at the interface is made. By using the Fourier integral transform (in perpendicular directions in each materials), the mixed boundary conditions and continuity conditions are transformed to a singular integral equation with generalized Cauchy kernel, the solution of which has been well studied, and classical methods are directly applicable here to obtain the closed form solution. The results are presented for a permeable crack under anti-plane shear loading and in-plane electric and magnetic loadings, as prescribed electric displacement and magnetic inductions or electric and magnetic fields. The results indicate that the magnetoelectroelastic field near the crack tip in the homogeneous PEMO- elastic ceramic is dominated by a traditional inverse square-root singularity, while the coupled field near the crack tip at the interface exhibits the singularity of the power law r--α , r being the distance from the interface crack tip and α depending on the material constants of a bimaterial. In particular, electric and magnetic fields have no singularity at the crack tip in a homogeneous solid, whereas they are singular around the interface crack tip. Numerical results are given graphically to show the effects of the material properties on the singularity order, field intensity factors and energy release rates. The results presented in this paper should have potential applications to the design of multilayered magnetoelectroelastic structures.

17

Analysis of mode I conducting crack in piezo-electro-magneto-elastic layer

Rogowski B.

International Journal of Applied Mechanics and Engineering

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2013

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Vol. 18, no. 1

153--176

EN

Within the theory of linear magnetoelectroelasticity, the fracture analysis of a magneto - electrically dielectric crack embedded in a magnetoelectroelastic layer is investigated. The prescribed displacement, electric potential and magnetic potential boundary conditions on the layer surfaces are adopted. Applying the Hankel transform technique, the boundary - value problem is reduced to solving three coupling Fredholm integral equations of second kind. These equations are solved exactly. The corresponding semi - permeable crack - face magnetoelectric boundary conditions are adopted and the electric displacement and magnetic induction of crack interior are obtained explicitly. This field inside the crack is dependent on the material properties, applied loadings, the dielectric permittivity and magnetic permeability of crack interior, and the ratio of the crack length and the layer thickness. Field intensity factors are obtained as explicit expressions.

18

Application of Rank Controlled Differential Quadrature Method for Solving an Infinite Steel Plate Cooling Problem

Żak P. L., Suchy J. S., Lelito J., Gracz B.

Archives of Foundry Engineering

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2013

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Vol. 13, iss. 3 spec.

201--206

EN

Rank Controlled Differential Quadrature method is a numerical method that allows to approximate the partial derivatives that appears in partial differential equations. Those equations with proper geometrical, physical, initial and boundary conditions make mathematical models of physical process. The heat transfer process is governed by Fourier–Kirchhoff equation, which is parabolic Partial Differential Equation. In this paper authors present the steel plate cooling problem. At the beginning of the process plate is heated up to 450 °C and is cooled to ambient temperature. The cooling of the plate is basic heat transfer problem. If the plates dimensions has proper proportions such problem may be described as one dimensional and solved exactly. The mathematical model and exact solution is given in the work. Authors apply the Rank Controlled Differential Quadrature to approximate derivatives in Fourier–Kirchhoff equation and in boundary conditions. After changing derivatives into quadrature formulation set of algebraic equations is obtained. Substituting thermo-physical parameters numerical model is obtained. The computer program was prepared to solve the problem numerically. Results of simulation are confronted with the exact ones. Error value at each time step as well as error value increase rate for examined numerical method is analyzed.

19

Fracture analysis of a mode I magneto-electrically conducting crack in piezo-electro-magneto-elastic medium

Rogowski B.

International Journal of Applied Mechanics and Engineering

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2012

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Vol. 17, no. 4

1267-1295

EN

Within the theory of linear magnetoelectroelasticity, the fracture analysis of a magneto electrically conducting crack embedded in a magneto electro elastic medium is investigated. The prescribed normal stress and two cases of electromagnetic boundary conditions applied at infinity are adopted. Applying the Hankel transform technique, the boundary value problem is reduced to solving three pairs of dual coupling integral equations. These equations are solved exactly. The corresponding semi permeable crack face magneto electric boundary conditions are adopted and the electric displacement and magnetic induction of the crack interior are obtained explicitly. This field inside the crack is dependent on the material properties, applied loadings and the dielectric permittivity and magnetic permeability of the crack interior. Field intensity factors are obtained as explicit expressions.

20

Flow analysis of viscoelastic fluid due to a shrinking sheet

Nandeppanavar M. M., Siddalingappa M. N., Kemparaju M. C.

International Journal of Applied Mechanics and Engineering

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2012

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Vol. 17, no. 1

249-257

EN

This work is concerned with the analysis of a two dimensional flow of a viscoelastic fluid (Walters' liquid B) induced by a shrinking sheet. The governing partial differential equations are first reduced to ordinary nonlinear differential equations by using the appropriate similarity transformation. An exponential solution is assumed to solve the considered boundary value problem. The effects of governing pertinent physical parameters on the flow characteristics are presented graphically and discussed.