Geometrically nonlinear composite shells with integrated piezoelectric layers

In order to show the significance of considering nonlinear deformations in piezoelectrical structures, a geometrical nonlinear shell element with integrated piezoelectric layers is introduced. The strain‐displacement relations are implemented using firstorder shear deformation theory with small strains but moderate rotations. The element has been tested on several benchmark problems and it can be concluded that the effect of geometrical nonlinearity is significant when the sensor properties of the piezoelectrical layers are predicted. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams considering surface energy effects

In this paper, the linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams are studied based on the Gurtin–Murdoch surface stress theory. Firstly, the constitutive equations of fractional viscoelasticity theory are considered, and based on the Gurtin–Murdoch model, stress components on the surface of the nanobeam are incorporated into the axial stress tensor. Afterward, using Hamilton's principle, equations governing the two-dimensional vibrations of fractional viscoelastic nanobeams are derived. Finally, two solution procedures are utilized to describe the time responses of nanobeams. In the first method, which is fully numerical, the generalized differential quadrature and finite difference methods are used to discretize the linear part of the governing equations in spatial and time domains. In the second method, which is semi-analytical, the Galerkin approach is first used to discretize nonlinear partial differential governing equations in the spatial domain, and the obtained set of fractional-order ordinary differential equations are then solved by the predictor–corrector method. The accuracy of the results for the linear and nonlinear vibrations of fractional viscoelastic nanobeams with different boundary conditions is shown. Also, by comparing obtained results for different values of some parameters such as viscoelasticity coefficient, order of fractional derivative and parameters of surface stress model, their effects on the frequency and damping of vibrations of the fractional viscoelastic nanobeams are investigated.     

A general nonlocal nonlinear model for buckling of nanobeams

This study presents a unified model for the nonlocal response of nanobeams in buckling and postbuckling states. The formulation is suitable for the classical Euler–Bernoulli, first-order Timoshenko, and higher-order shear deformation beam theories. The small-scale effect is modeled according to the nonlocal elasticity theory of Eringen. The equations of equilibrium are obtained using the principle of virtual work. The stress resultants are developed taking into account the nonlocal effect. Analytical solutions for the critical buckling load and the amplitude of the static nonlinear response in the postbuckling state are obtained. It is found out that as the nonlocal parameter increases, the critical buckling load reduces and the amplitude of buckling increases. Numerical results showing variation of the critical buckling load and the amplitude of buckling with the nonlocal parameter and the length-to-height ratio for simply supported and clamped–clamped nanobeams are presented.     

Geometrically nonlinear bending analysis of laminated composite plate

In this work, a transverse bending of shear deformable laminated composite plates in Green–Lagrange sense accounting for the transverse shear and large rotations are presented. Governing equations are developed in the framework of higher order shear deformation theory. All higher order terms arising from nonlinear strain–displacement relations are included in the formulation. The present plate theory satisfies zero transverse shear strains conditions at the top and bottom surfaces of the plate in von-Karman sense. A C⁰ isoparametric finite element is developed for the present nonlinear model. Numerical results for the laminated composite plates of orthotropic materials with different system parameters and boundary conditions are found out. The results are also compared with those available in the literature. Some new results with different parameters are also presented.     

Buckling and vibration analysis of nonlocal axially functionally graded nanobeams based on Hencky-bar chain model

Buckling and free vibration analyses of nonlocal axially functionally graded Euler nanobeams is the main objective of this paper. Due to its simplicity, the Eringen's differential constitutive model is adopted for describing the nonlocal size dependency of nanostructure beam. The nonlocal equilibrium equation is derived using the principle of the minimum potential energy principle, and discretized by using the link-spring model known in literature as Hencky bar-chain model. The general applicability of the proposed approach allows analyses of functional graded microbeams without any restriction on variability, boundary and loading conditions. A comparison with results available in the literature shows the reliability of the method.     

Uniform stabilization of a nonlinear structural acoustic model with a Timoshenko beam interface

This paper is concerned with a nonlinear model which describes the interaction of sound and elastic waves in a two‐dimensional acoustic chamber in which one flat ‘wall’, the interface, is flexible. The composite dynamics of the structural acoustic model is described by the linearized equations for a gas defined on the interior of the chamber and the nonlinear Timoshenko beam equations on the interface. Uniform stability of the energy associated with the interactive system of partial differential equations is achieved by incorporating a nonlinear feedback boundary damping scheme in the equations for the gas and the beam. Copyright © 2006 John Wiley & Sons, Ltd.     

Geometrically nonlinear shell finite element based on the geometry of a planar curve

An engineering approach for constructing a curved triangular finite element of a thin shell is considered. The approach is based on the assumption that the triangle sides are planar nearly circular curves before and after deformation. A geometrically nonlinear formulation of a triangular finite element of a thin Kirchhoff–Love shell is given. The predictive capabilities of the element are tested using benchmark problems of nonlinear deformation of elastic plates and shells.     

Geometrically nonlinear static and dynamic analysis of functionally graded skew plates

The present paper deals with nonlinear static and dynamic behavior of functionally graded skew plates. The equations of motion are derived using higher order shear deformation theory in conjunction with von-Karman’s nonlinear kinematics. The physical domain is mapped into computational domain using linear mapping and chain rule of differentiation. The spatial and temporal discretization is based on fast converging finite double Chebyshev series and Houbolt’s method. Quadratic extrapolation technique is employed to linearize the governing nonlinear equations. The spatial and temporal convergence and validation studies have been carried out to establish the efficacy of the present solution methodology. In case of dynamic analysis, the results are obtained for uniform step, sine, half sine, triangular and exponential type of loadings. The effect of volume fraction index, skew angle and boundary conditions on nonlinear displacement and moment response are presented.     

Stress analysis near holes in shells of revolution: Geometrically nonlinear problem

An approach is proposed to numerical analysis of the stress state of shells of revolution in the geometrically nonlinear setting. It is based on a combination of linearization and discrete orthogonalization methods. Stress concentration is analyzed for a flexible spherical shell with a central hole, depending on load intensity, hole size, and the boundary conditions on the contours in the precritical strain region.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 97–101, 1985.     

Parameter analysis based on stochastic model for differential evolution algorithm

In this paper, a stochastic model is used to describe and analyze the evolution process of differential evolution (DE) for numerical optimization. With the model, it illustrates how the probability distribution of the whole population is changed by mutation, selection and crossover operations. Based on the theoretical analysis, some guidelines about the parameter setting for DE are provided. In addition, numerical simulations are carried out to verify the conclusions drawn from model analysis.     

Geometrically nonlinear analysis with a 4-node membrane element formulated by the quadrilateral area coordinate method

Recently, a 4-node quadrilateral membrane element AGQ6-I, has been successfully developed for analysis of linear plane problems. Since this model is formulated by the quadrilateral area coordinate method (QACM), a new natural coordinate system for developing quadrilateral finite element models, it is much less sensitive to mesh distortion than other 4-node isoparametric elements and free of various locking problems that arise from irregular mesh geometries. In order to extend these advantages of QACM to nonlinear applications, the total Lagrangian (TL) formulations of element AGQ6-I was established in this paper, which is also the first time that a plane QACM element being applied in the implicit geometrically nonlinear analysis. Numerical examples of geometrically nonlinear analysis show that the presented formulations can prevent loss of accuracy in severely distorted meshes, and therefore, are superior to those of other 4-node isoparametric elements. The efficiency of QACM for developing simple, effective and reliable serendipity plane membrane elements in geometrically nonlinear analysis is demonstrated clearly.     

Geometrically nonlinear dilation of an orthotropic plane with an elliptical hole

In a geometrically nonlinear formulation we obtain expressions for the stresses and displacements on the edge of an elliptical hole in an orthotropic plane dilated by a strain at infinity in the direction of the principal axes of anisotropy. We exhibit the dependence of the concentration of stresses on the magnitude of the load acting on the plane. We prove that the stresses are finite at the ends of the crack and we compute their values. We confirm the need to take account of the geometric nonlinearity of the problem for low-modulus orthotropic materials.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 90–96.     

Surface effects on nonlinear free vibration of functionally graded nanobeams using nonlocal elasticity

In this paper, to consider all surface effects including surface elasticity, surface stress, and surface density, on the nonlinear free vibration analysis of simply-supported functionally graded Euler–Bernoulli nanobeams using nonlocal elasticity theory, the balance conditions between FG nanobeam bulk and its surfaces are considered to be satisfied assuming a cubic variation for the component of the normal stress through the FG nanobeam thickness. The nonlinear governing equation includes the von Kármán geometric nonlinearity and the material properties change continuously through the thickness of the FG nanobeam according to a power-law distribution of the volume fraction of the constituents. The multiple scale method is employed as an analytical solution for the nonlinear governing equation to obtain the nonlinear natural frequencies of FG nanobeams. The effect of the gradient index, the nanobeam length, thickness to length ratio, mode number, amplitude of deflection to radius of gyration ratio and nonlocal parameter on the frequency ratios of FG nanobeams is investigated.     

Oscillation of second-order nonlinear differential equations with nonlinear damping

This paper is concerned with the oscillation of a class of general type second-order differential equations with nonlinear damping terms. Several new oscillation criteria are established for such a class of differential equations under quite general assumptions. Examples are also given to illustrate the results.     

Existence of periodic solutions of nonlinear systems of differential equations with impulse effect

Sufficient conditions for the existence of periodic solutions of nonlinear systems of differential equations with impulses in a canonic domain have been found in the paper.     

Analysis of nonlinear fractional partial differential equations with the homotopy analysis method

In this paper, the time fractional partial differential equations are investigated by means of the homotopy analysis method. This technique is extended to study the partial differential equations of fractal order for the first time. The accurate series solutions are obtained. This indicates the validity and great potential of the homotopy analysis method for solving nonlinear fractional partial differential equations.     

Bifurcation analysis of an epidemic model with nonlinear incidence

In this paper, we consider an epidemic model with the nonlinear incidence of a sigmoidal function. By mathematical analysis, it is shown that the model exhibits the bistability and undergoes the Hopf bifurcation and the Bogdanov-Takens bifurcation. By numerical simulations, it is found that the incidence rate can induce multiple limit cycles, and a little change of the parameter could lead to quite different bifurcation structures.     

A direct asymptotic analysis on a nonlinear model with thin layers

We derive an asymptotic model from a nonlinear model with thin layers letting the thickness goes to 0. Usually these problems are written in a functional minimisation form and the asymptotic analysis is performed using epiconvergence. Here we give a direct proof by extending the method used on a linear model by E. Sanchez-Palencia in [J. Math. Pures et Appl., 53, 1974, 711–740]. Finally we give an application to a water transfert project studied by the Yangtse River Scientific Research Institute in China.

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Sunto Deriviamo un modello asintotico da un modello nonlineare a strati sottili facendo tendere lo spessore a zero. Usualmente questi problemi sono scritti in forma di minimizzazione funzionale e l'analisi asintotica è trattata usando epiconvergenza. Qui diamo una prova dirtta estendendo il metodo usato su un modello lineare da E. Sanchez-Palencia in [J. Math. Pures et Appl., 53, 1974, 711–740]. Infine, diamo un'applicazione al progetto di trasferimento dell'acqua studiato nell'istituto cinese Yangtse River Scientific Research Institute.

     

The nonlinear analysis on a discrete host-parasitoid model with pesticidal interference

A host-parasitoid model with a pesticidal interference is investigated. The global bifurcations of feasible sets and domains, caused by pesticide-related mortality, are given in the light of some recent results on iterated maps with vanishing denominator giving rise to non-classical singularities: a non-definition line, a focal point and a prefocal line. In particular, we give an answer to one of the open problems proposed by Kocic and Ladas.