Nonlinear analysis of buckling,free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment

Employing Euler–Bernoulli beam theory and the physical neutral surface concept, the nonlinear governing equation for the functionally graded material beam with two clamped ends and surface-bonded piezoelectric actuators is derived by the Hamilton’s principle. The thermo-piezoelectric buckling, nonlinear free vibration and dynamic stability for the piezoelectric functionally graded beams, subjected to one-dimensional steady heat conduction in the thickness direction, are studied. The critical buckling loads for the beam are obtained by the existing methods in the analysis of thermo-piezoelectric buckling. The Galerkin’s procedure and elliptic function are adopted to obtain the analytical solution of the nonlinear free vibration, and the incremental harmonic balance method is applied to obtain the principle unstable regions of the piezoelectric functionally graded beam. In the numerical examples, the good agreements between the present results and existing solutions verify the validity and accuracy of the present analysis and solving method. Simultaneously, validation of the results achieved by rule of mixture against those obtained via the Mori–Tanaka scheme is carried out, and excellent agreements are reported. The effects of the thermal load, electric load, and thermal properties of the constituent materials on the thermo-piezoelectric buckling, nonlinear free vibration, and dynamic stability of the piezoelectric functionally graded beam are discussed, and some meaningful conclusions have been drawn.     

Control of sound and vibration for cylindrical shells by utilizing a periodic structure of functionally graded material

A periodic shell made of functionally graded material (FGM) is proposed in this Letter. Wave propagation and vibration transmission in the FGM periodic shell for different circumferential modes are investigated. By illustrating the dynamical behavior of the periodic FGM shell within the pass/stop band frequency ranges, the mechanism of wave propagation and vibration transmission in the shell are illuminated. Moreover, the suppression characteristics of structure-borne sound in the internal field of the shell, either within the stop or pass band frequency ranges, are studied.     

An improved isogeometrical analysis approach to functionally graded plane elasticity problems

The isogeometric analysis method is extended for addressing the plane elasticity problems with functionally graded materials. The proposed method which employs an improved form of the isogeometric analysis approach allows gradation of material properties through the patches and is given the name Generalized Iso-Geometrical Analysis (GIGA). The gradations of materials, which are considered as imaginary surfaces over the computational domain, are defined in a fully isoparametric formulation by using the same NURBS basis functions employed for the construction of the geometry and the approximation of the solution. The basic concept of the developed approach is concisely explained and its relation to the standard isogeometric analysis method is pointed out. It is shown that the difficulties encountered in the finite element analysis of the functionally graded materials are alleviated to a large degree by employing the mentioned method. Different numerical examples are presented and compared with available analytical solutions as well as the conventional and graded finite element methods to demonstrate the performance and accuracy of the proposed approach. The presented procedure can also be employed for solving other partial differential equations with non-constant coefficients.     

Bending solutions of FGM Timoshenko beams from those of the homogenous Euler–Bernoulli beams

By using mathematical similarity and load equivalence between the governing equations, bending solutions of FGM Timoshenko beams are derived analytically in terms of the homogenous Euler–Bernoulli beams. The deflection, rotational angle, bending moment and shear force of FGM Timoshenko beams are expressed in terms of the deflection of the corresponding homogenous Euler–Bernoulli beams with the same geometry, the same loadings and end constraints. Consequently, solutions of bending of the FGM Timoshenko beams are simplified as the calculation of the transition coefficients which can be easily determined by the variation law of the gradient of the material properties and the geometry of the beams if the solutions of corresponding Euler–Bernoulli beam are known. As examples, solutions are given for the FGM Timoshenko beams under S–S, C–C, C–F and C–S end constraints and arbitrary transverse loadings to illustrate the use of this approach. These analytical solutions can be as benchmarks in the further investigations of behaviors of FGM beams.     

Large-amplitude free vibrations of functionally graded beams by means of a finite element formulation

The large-amplitude free vibration analysis of functionally graded beams is investigated by means of a finite element formulation. The Von-Karman type nonlinear strain–displacement relationships are employed where the ends of the beam are constrained to move axially. The effects of the transverse shear deformation and rotary inertia are included based upon the Timoshenko beam theory. The material properties are assumed to be graded in the thickness direction according to the power-law distribution. A statically exact beam element which devoid the shear locking effect with displacement fields based on the first order shear deformation theory is used to study the geometric nonlinear effects on the vibrational characteristics of functionally graded beams. The finite element method is employed to discretize the nonlinear governing equations, which are then solved by the direct numerical integration technique in order to obtain the nonlinear vibration frequencies of functionally graded beams with different boundary conditions. The influences of power-law exponent, vibration amplitude, beam geometrical parameters and end supports on the free vibration frequencies are studied. The present numerical results compare very well with the results available from the literature where possible. Some new results for the nonlinear natural frequencies are presented in both tabular and graphical forms which can be used for future references.     

High-order tetrahedral finite elements applied to large deformation analysis of functionally graded rubber-like materials

In this paper, high-order tetrahedral finite elements are employed to analyze structures and solids composed of functionally graded rubber-like materials under finite displacements, finite strains, statically applied forces and isothermal conditions. In order to do so, the following concepts are used: geometrically nonlinear analysis, Green–Lagrange strain tensor, second Piola–Kirchhoff stress tensor, hyperelastic constitutive relations, isoparametric solid tetrahedral finite elements of any order of approximation, and functionally graded materials. The equilibrium of the body is achieved via the Principle of the Stationary Total Potential Energy. The elements are fully integrated via Gaussian quadratures, and the resultant processing time is reduced by means of parallel techniques. To solve the nonlinear system of equations, the Newton–Raphson iterative procedure is employed.The proposed formulation is validated by benchmark problems such as: the Cook’s membrane and the thick cylinder. Other interesting simulation, the Cook’s block is proposed in order to evaluate high strain gradient situations. The results show that, in the context of the present study, locking-free behavior is obtained with simple mesh refinement.     

A CQM-based BEM for transient heat conduction problems in homogeneous materials and FGMs

A method for numerical solution of time-domain boundary integral formulations of transient problems governed by the heat equation is presented. The heat conduction problem is analyzed considering homogeneous and non-homogeneous media. In the case of the non-homogeneous media, the conductor material is assumed to be a functionally graded material, i.e., the material properties vary spatially according to known smooth functions. For some specific spatial variations of the material properties, the fundamental solution and the boundary integral equation of the problem are obtained thanks to a change of variables that transforms the original problem to the standard heat conduction problem for homogeneous materials. For the treatment of time-dependent terms, the convolution quadrature method is adopted to approximate numerically the integral equation of the time-domain boundary element method. In the case that the responses are required at a large number of interior points, the convolution performed to calculate them is very time consuming. It is shown that the discrete convolution of the proposed formulation can be computed by means of the fast Fourier transform technique, which considerably reduces the computational complexity. Results for some transient heat conduction examples are presented to validate the numerical techniques studied.     

Transient thermal stress analysis of a functionally graded magneto-electro-thermoelastic strip due to nonuniform surface heating

This article is concerned with the theoretical analysis of the functionally graded magneto-electro-thermoelastic strip due to unsteady and nonuniform surface heating in the width direction. We analyze the transient thermal stress problem for a functionally graded strip constructed of the anisotropic and linear magneto-electro-thermoelastic materials using a laminated composite mode as one of theoretical approximation. The transient two-dimensional temperature is analyzed by the methods of Laplace and finite sine transformations. We obtain the solution for the simply supported and functionally graded magneto-electro-thermoelastic strip under a plane strain state. As an illustration, we carried out numerical calculations for a functionally graded strip composed of piezoelectric BaTiO₃ and magnetostrictive CoFe₂O₄, and examined the behaviors in the transient state for temperature change, stress, electric potential and magnetic potential distributions. Furthermore, the effects of the nonhomogeneity of material on the stresses, electric potential, and magnetic potential are investigated.     

Surface waves in an exponentially graded, general anisotropic elastic material under the influence of gravity

In a recent paper Destrade [1] studied surface waves in an exponentially graded orthotropic elastic material. He showed that the quartic equation for the Stroh eigenvalue p is, after properly modified, a quadratic equation in p² with real coefficients. He also showed that the displacement and the stress decay at different rates with the depth x₂ of the half-space. Vinh and Seriani [2] considered the same problem and added the influence of gravity on surface waves. In this paper we generalize the problem to exponentially graded general anisotropic elastic materials. We prove that the coefficients of the sextic equation for p remain real and that the different decay rates for the displacement and the stress hold also for general anisotropic materials. A surface wave exists in the graded material under the influence of gravity if a surface wave can propagate in the homogeneous material without the influence of gravity in which the material parameters are taken at the surface of the graded half-space. As the wave number k → ∞, the surface wave speed approaches the surface wave speed for the homogeneous material. A new matrix differential equation for surface waves in an arbitrarily graded anisotropic elastic material under the influence of gravity is presented. Finally we discuss the existence of one-component surface waves in the exponentially graded anisotropic elastic material with or without the influence of gravity.