A new approach to the analysis of shells,plates and membranes with finite deflections

This paper presents a new perturbation method of analysis applicable to a class of geometrically non-linear problems of shells, plates, and membranes with translationally restrained edges. The perturbation parameter is a linear function of Poisson's ratio. The convergence of successive perturbations (i.e., approximations) is independent of the magnitudes of deflections. The method also offers a rational explanation of the efficacy of Berger's approximate equations, thus placing Berger's method on a firmer foundation while at the same time weakening his hypothesis of vanishing second membrane strain invariant in the strain energy integral. Several solutions and results are obtained for the purposes of illustration and discussion. Whenever possible, calculated values are compared with results obtained by other means.     

Post-buckling analysis of slender elastic vertical rods subjected to terminal forces and self-weight

This article presents the behavior of slender elastic rods subjected to axial terminal forces and self-weight. The mathematical formulation is presented, a solution is sought for a double-hinged boundary condition and the analysis is carried out for different values of non-dimensional weight. The formulation derives from geometrical compatibility, equilibrium of forces and moments and constitutive relations yielding a set of six first order non-linear ordinary differential equations with boundary conditions specified at both ends, which characterizes a complex two-point boundary value problem. Furthermore, a perturbation method is used to find the critical buckling loads and initial post-buckling solutions. A numerical integration scheme based on a three parameter shooting method is employed in the post-buckling solutions.     

Geometrically non-linear free and forced vibrations of simply supported circular cylindrical shells: A semi-analytical approach

The non-linear free and forced vibrations of simply supported thin circular cylindrical shells are investigated using Lagrange's equations and an improved transverse displacement expansion. The purpose of this approach was to provide engineers and designers with an easy method for determining the shell non-linear mode shapes, with their corresponding amplitude dependent non-linear frequencies. The Donnell non-linear shell theory has been used and the flexural deformations at large vibration amplitudes have been taken into account. The transverse displacement expansion has been made using two terms including both the driven and the axisymmetric modes, and satisfying the simply supported boundary conditions. The non-linear dynamic variational problem obtained by applying Lagrange's equations was then transformed into a static case by adopting the harmonic balance method. Minimisation of the energy functional with respect to the basic function contribution coefficients has led to a simple non-linear multi-modal equation, the solution of which gives in the case of a single mode assumption an expression for the non-linear frequencies which is much simpler than that derived from the non-linear partial differential equation obtained previously by several authors. Quantitative results based on the present approach have been computed and compared with experimental data. The good agreement found was very satisfactory, in comparison with previous old and recent theoretical approaches, based on sophisticated numerical methods, such as the finite element method (FEM), the method of normal forms (MNF), and analytical methods, such as the perturbation method.     

Hydromagnetic flow of a dusty fluid over a stretching sheet

Analysis of hydromagnetic flow of a dusty fluid over a stretching sheet is carried out with a view to throw adequate light on the effects of fluid-particle interaction, particle loading, and suction on the flow characteristics. The equations of motion are reduced to coupled non-linear ordinary differential equations by similarity transformations. These coupled non-linear ordinary differential equations are solved numerically on an IBM 4381 with double precession, using a variable order, variable step-size finite-difference method. The numerical solutions are compared with their approximate solutions, obtained by a perturbation technique. For small values of β the exact (numerical) solution is in close agreement with that of the analytical (approximate) solution. It is observed that, even in the presence of a transverse magnetic field and suction, the transverse velocity of both the fluid and particle G phases decreases with an increase in the fluid-particle interaction parameter, β, or the particle-loading parameter, k. Moreover, the particle density is maximum at the surface of the stretching sheet, and the shearing stress increases with an increase in β or k.     

MHD free convective flow and mass transfer over a stretching sheet with chemical reaction

The effect of chemical reaction on free convective flow and mass transfer of a viscous, incompressible and electrically conducting fluid over a stretching surface is investigated in the presence of a constant transverse magnetic field. The non-linear boundary layer equations with the boundary conditions are transferred by a similarity transformation into a system of non-linear ordinary differential equations with the appropriate boundary conditions. Furthermore, the similarity equations are solved numerically by using a fourth order Runge-Kutta scheme with the shooting method. Numerical results of the skin friction coefficient, the local Nusselt number Nu, the local Sherwood number Sh, as will as the velocity, temperature and concentration profiles are presented for gases with a Prandtl number of 0.71 for various values of chemical reaction parameter, order of reaction, magnetic parameter and Schmidt number.     

Initial thermo-mechanical post-buckling of beams with temperature-dependent physical properties

The objective of this work is to analyze the elastic buckling and initial post-buckling behavior of slender beams subjected to uniform heating. The beams are assumed to be double-hinged with fixed ends, preventing thermal expansion. Consequently, destabilizing compressive forces arise that may lead to beam buckling. When the temperature is further increased, the beam experiences finite displacements, with the result that the analysis is geometrically non-linear. The modulus of elasticity and the thermal induced strain, key material properties for this problem, are temperature-dependent. Thus, the coefficients of the governing equations are not constant. This suggests the physical non-linearity of the mathematical model. Hence, the analysis is geometrically and physically non-linear. The analysis is sensitive to the beam initial temperature, as the thermal strain is a function of the initial and final temperatures. The material is considered to be linear elastic, and consequently viscoelastic and plastic effects are not taken into account. Furthermore, the beam cross-section properties are assumed to be constant, which is consistent with the small strain formulation. A perturbation method is applied to the governing non-linear differential equations so that the initial post-buckling behavior may be analytically determined when temperatures above the critical temperature are applied to the beam. To illustrate the application of the formulation we present a case study for the aluminum 7075-T6 alloy, a material commonly used in aerospace and naval industries. Nonetheless, it is expected similar behavior for other metallic materials. The curves that define the variation of the modulus of elasticity, the thermal strain and the yield stress with temperature are considered in our analysis. The change in length, reaction forces at the supports and geometric configurations are obtained as a function of temperature and the beam slenderness ratio. The critical buckling loads and temperatures and the initial post-buckling analysis are also calculated in the context of the temperature-independent physical properties. Our results emphasize the importance of modeling the material's non-linearity if accuracy is required. However, from a practical application point of view results are acceptable if temperature-independent physical properties are employed, especially for large slenderness ratios.     

MHD free convective flow past semi-infinite vertical permeable wall

In this paper,the basic equations governing the flow and heat transfer of an incompressible viscous and electrically conducting fluid past a semi-infinite vertical permeable plate in the form of partial differential equations are reduced to a set of non-linear ordinary differential equations by applying a suitable similarity transformation.Approximate solutions of the transformed equations are obtained by employing the perturbation method for two cases,i.e.,small and large values of the suction parameter.From the numerical evaluations of the solution,it can be seen that the velocity field at any point decreases as the values of the magnetic and suction parameters increase.The effect of the magnetic parameter is to increase the thermal boundary layer.It is also found that the velocity and temperature fields decrease with the increase in the sink parameter.     

Planar non-linear free vibrations of an elastic cable

Continuum non-linear equations of free motion of a heavy elastic cable about a deformed initial configuration are developed. Referring to an assumed mode technique one ordinary equation for the cable planar motion is obtained via a Galerkin procedure, an approximate solution of which is pursued through a perturbation method. Suitable nondimensional results are presented for the vibrations in the first symmetric mode with different values of the cable properties. Which procedure is the proper one to account consistently for the non-linear kinematical relations of the cable in one ordinary equation of motion is discussed.     

Stability analysis of viscoelastic thin shallow hyperbolic paraboloid shells

The stability of linearly viscoelastic flexible shallow hyperbolic paraboloid shell is analysed under transverse load. Allowances are made for geometrical nonlinearity and initial imperfections of the surface shape. By application of the method of finite differences with respect to geometrical variables and the method of differentiation with respect to a parameter (time) the solution for the system of equilibrium non-linear integro-differential equations is reduced to Cauchys problem which can be solved numerically. The critical time was shown to depend on the load, curvature, initial imperfections and edge elements compressibility. Critical loads for an outlying time moment are determined.     

Non-linear vibrations of suspension bridges with external excitation

Non-linear coupled vertical and torsional vibrations of suspension bridges are investigated. Method of Multiple Scales, a perturbation technique, is applied to the equations to find approximate analytical solutions. The equations are not discretized as usually done, rather the perturbation method is applied directly to the partial differential equations. Free and forced vibrations with damping are investigated in detail. Amplitude and phase modulation equations are obtained. The dependence of non-linear frequency on amplitude is described. Steady-state solutions are analyzed. Frequency-response equation is derived and the jump phenomenon in the frequency-response curves resulting from non-linearity is considered. Effects of initial amplitude and phase values, amplitude of excitation, and damping coefficient on modal amplitudes, are determined.     

Nonlinear three-dimensional stretched flow of an Oldroyd-B fluid with convective condition,thermal radiation,and mixed convection

The effect of non-linear convection in a laminar three-dimensional Oldroyd-B fluid flow is addressed. The heat transfer phenomenon is explored by considering the non-linear thermal radiation and heat generation/absorption. The boundary layer assumptions are taken into account to govern the mathematical model of the flow analysis. Some suitable similarity variables are introduced to transform the partial differential equations into ordinary differential systems. The Runge-Kutta-Fehlberg fourth-and fifth-order techniques with the shooting method are used to obtain the solutions of the dimensionless velocities and temperature. The effects of various physical parameters on the fluid velocities and temperature are plotted and examined. A comparison with the exact and homotopy perturbation solutions is made for the viscous fluid case, and an excellent match is noted. The numerical values of the wall shear stresses and the heat transfer rate at the wall are tabulated and investigated. The enhancement in the values of the Deborah number shows a reverse behavior on the liquid velocities. The results show that the temperature and the thermal boundary layer are reduced when the nonlinear convection parameter increases. The values of the Nusselt number are higher in the non-linear radiation situation than those in the linear radiation situation.     

Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory

In this study, non-linear free vibration of micro-plates based on strain gradient elasticity theory is investigated. A general form of Mindlin’s first-strain gradient elasticity theory is employed to obtain a general Kirchhoff micro-plate formulation. The von Karman strain tensor is used to capture the geometric non-linearity. The governing equations of motion and boundary conditions are obtained in a variational framework. The Homotopy analysis method is employed to obtain an accurate analytical expression for the non-linear natural frequency of vibration. For some specific values of the gradient-based material parameters, the general plate formulation can be reduced to those based on some special forms of strain gradient elasticity theory. Accordingly, three different micro-plate formulations are introduced, which are based on three special strain gradient elasticity theories. It is found that both geometric non-linearity and size effect increase the natural frequency of vibration. In a micro-plate having a thickness comparable with the material length scale parameter, the strain gradient effect on increasing the non-linear natural frequency is higher than that of the geometric non-linearity. By increasing the plate thickness, the strain gradient effect decreases or even diminishes. In this case, geometric non-linearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for micro-plates with some specific thickness to length scale parameter ratios, both geometric non-linearity and size effect have significant role on increasing the frequency of non-linear vibration.     

The dynamic behavior of MEMS arch resonators actuated electrically

In this paper, we investigate the dynamic behavior of clamped-clamped micromachined arches when actuated by a small DC electrostatic load superimposed to an AC harmonic load. A Galerkin-based reduced-order model is derived and utilized to simulate the static behavior and the eigenvalue problem under the DC load actuation. The natural frequencies and mode shapes of the arch are calculated for various values of DC voltages and initial rises. In addition, the dynamic behavior of the arch under the actuation of a DC load superimposed to an AC harmonic load is investigated. A perturbation method, the method of multiple scales, is used to obtain analytically the forced vibration response of the arch due to DC and small AC loads. Results of the perturbation method are compared with those obtained by numerically integrating the reduced-order model equations. The non-linear resonance frequency and the effective non-linearity of the arch are calculated as a function of the initial rise and the DC and AC loads. The results show locally softening-type behavior for the resonance frequency for all DC and AC loads as well as the initial rise of the arch.     

Periodic and quasiperiodic motion for complex non-linear systems

A damped complex non-linear system corresponding to two coupled non-linear oscillators with a periodic damping force is investigated by an asymptotic perturbation method based on Fourier expansion and time rescaling. Four coupled equations for the amplitude and the phase of solutions are derived. Phase-locked solutions with period equal to the damping force period are possible only if the oscillators amplitudes are equal. On the contrary, if the oscillators amplitudes are different, periodic solutions exist only with a period different from the damping force period. These solutions are stable only for perturbations that conserve the phase difference and the square amplitude sum of the oscillators. Energy considerations are used in order to study existence and characteristics of quasiperiodic motion. We demonstrate that modulated motion can be also obtained for appropriate values of the detuning parameter and in this case an approximate analytic solution is easily constructed. If the detuning parameter decreases the modulation period increases and then diverges, an infinite-period bifurcation occurs and the resulting motion becomes unbounded. Analytic approximate solutions are checked by numerical integration.     

Parametric resonance in non-linear elastodynamics

We show that finite amplitude shearing motions superimposed on an unsteady simple extension are admissible in any incompressible isotropic elastic material. We show that the determining equations for these shearing motions admit a general reduction to a system of ordinary differential equations (ODEs) in the remarkable case of generalized circularly polarized transverse waves. When these waves are standing and the underlying unsteady simple extension is composed of a harmonic perturbation of a static stretch it is possible to reduce the determining ODEs to linear or non-linear Mathieu equations. We use this property for a detailed study of the phenomenon of parametric resonance in non-linear elastodynamics.     

Localization of two-dimensional non-linear strain waves in a plate

The influence of an external medium on the evolution of two-dimensional long non-linear strain waves in an elastic plate is studied. The governing non-linear equations for longitudinal and shear waves are obtained. A threshold value of the external medium parameter is found that separates the existence of either one-dimensional (or plane) localized strain wave or two-dimensional localized strain wave. A considerable increase in the amplitude of the wave is found during the formation of the two-dimensional localized strain wave from an arbitrary initial pulse.     

RESEARCH OF VISCO-ELASTIC TYPE Ⅱ RUPTURE WITH EXCITING AND ATTENUATION PROCESS

With non-linear Rayleigh damping formula we describe the exciting process when the rupture velocity is low and the attenuation process when the rupture velocity reaches a certain high value. Assuming the medium of the earth crust is homogeneous and isotropic linear Voigt viscoelastic body, with small parameter perturbation method to deduce the non-linear governing partial differential equations into a system of asymptotic linear ones, we solve them by means of generalized fourier series with moving coordinates as its variables, thus transform them into non-homogeneous mathieu equations. At last Mathieu equations are solved by WKBJ method.     

Instabilities in bimetallic layers

The analysis of the thickness fluctuations in the extrusion process of bimetallic tubes has motivated the present work. For an orthotropic, incrementally-linear solid, the possibility of non-uniform solutions near the bimaterial interface is investigated by assuming an initial perturbation along the interface. The bifurcation equation for the problem is established and firstly solved numerically to obtain a critical strain in terms of a corresponding wavenumber. Then the bifurcation equation is solved for strains above the critical strain in order to analyse the growth of the perturbation, by means of an instability parameter introduced in the analysis. A set of values for several selected parameters of the process and three constitutive equations are considered in the computations. Their influence in the stability of the process is discussed in detail.     

High-order methods for differential equations with large first-derivative terms

A method is developed to solve elliptic singular perturbation problems. Examples are presented in one and two dimensions for both linear and non-linear problems. In particular, examples are presented for fluid flow problems with boundary layers. In the one-dimensional case an approximating equation is developed using just three points. The method first presented is a fourth-order approximation but is extended to become a higher-order method. Results are included for the fourth-, sixth-, eighth- and tenth-order methods. The results are first compared with results found by Segal in an article about elliptic singular perturbation problems. The elliptic singular perturbation problems are compared with a method by Il'in and also with central and backward difference schemes from Segal's article. There was only one case where the results in Segal's paper were as accurate as the results presented in this paper. However, in this case the method used by Segal did not give accurate values for a second problem presented. The results are also compared with results given by Spalding and by Christie. The method of this paper was also tested on the solution of some non-linear diffusion equations with concentration-dependent diffusion coefficients. The results were superior to results presented by Lee and by Schultz. Finally, the method is extended to several two-dimensional problems. The method developed in this paper is accurate, easy to use and can be generalized to other problems.