Wave propagation analysis of porous functionally graded piezoelectric nanoplates with a visco-Pasternak foundation

This study investigates the size-dependent wave propagation behaviors under the thermoelectric loads of porous functionally graded piezoelectric(FGP) nanoplates deposited in a viscoelastic foundation. It is assumed that(i) the material parameters of the nanoplates obey a power-law variation in thickness and(ii) the uniform porosity exists in the nanoplates. The combined effects of viscoelasticity and shear deformation are considered by using the Kelvin-Voigt viscoelastic model and the refined hi...     

Analysis of wave propagation in a functionally graded nanobeam resting on visco-Pasternak's foundation

Wave propagation analysis for a functionally graded nanobeam with rectangular cross-section resting on visco-Pasternak's foundation is studied in this paper. Timoshenko's beam model and nonlocal elasticity theory are employed for formulation of the problem. The equations of motion are derived using Hamilton's principals by calculating kinetic energy, strain energy and work due to viscoelastic foundation. The effects of various parameters such as wavenumber, non-homogeneous index, nonlocal parameter and three parameters of foundation are performed on the phase velocity of the nanobeam. The obtained results indicate that some parameters such as non-homogeneous index, nonlocal parameter and wavenumber have significant effect on the response of the system.     

Recursive geometric integrators for wave propagation in a functionally graded multilayered elastic medium

The differential equations governing transfer and stiffness matrices and acoustic impedance for a functionally graded generally anisotropic magneto-electro-elastic medium have been obtained. It is shown that the transfer matrix satisfies a linear 1st order matrix differential equation, while the stiffness matrix satisfies a nonlinear Riccati equation. For a thin nonhomogeneous layer, approximate solutions with different levels of accuracy have been formulated in the form of a transfer matrix using a geometrical integration in the form of a Magnus expansion. This integration method preserves qualitative features of the exact solution of the differential equation, in particular energy conservation. The wave propagation solution for a thick layer or a multilayered structure of inhomogeneous layers is obtained recursively from the thin layer solutions. Since the transfer matrix solution becomes computationally unstable with increase of frequency or layer thickness, we reformulate the solution in the form of a stable stiffness-matrix solution which is obtained from the relation of the stiffness matrices to the transfer matrices. Using an efficient recursive algorithm, the stiffness matrices of the thin nonhomogeneous layer are combined to obtain the total stiffness matrix for an arbitrary functionally graded multilayered system. It is shown that the round-off error for the stiffness-matrix recursive algorithm is higher than that for the transfer matrices. To optimize the recursive procedure, a computationally stable hybrid method is proposed which first starts the recursive computation with the transfer matrices and then, as the thickness increases, transits to the stiffness matrix recursive algorithm. Numerical results show this solution to be stable and efficient. As an application example, we calculate the surface wave velocity dispersion for a functionally graded coating on a semispace.     

Wave propagation across a finite heterogeneous interphase modeled as an interface with material properties

Mechanical problems involving an interphase between two well-defined, and eventually different, materials are of interest. The aim of this paper is to present a simplified model that, for low frequency regime, is appropriate for this situation: an interface model with elastic and inertial properties. We present, together with the equations of motion, an identification procedure that is valid for any mass density profile along the thickness of the interphase. For evaluating the accuracy of the model, computations of the reflection coefficients in some relevant cases are shown. Besides, a finite element method is used as a benchmark for both the high and low frequency regimes. It is worth to be noted that the numerical test has been inspired by the problem of the interphase that is formed at the bone-implant boundary.     

Free vibration characteristics of sectioned unidirectional/bidirectional functionally graded material cantilever beams based on finite element analysis

Advancements in manufacturing technology, including the rapid development of additive manufacturing (AM), allow the fabrication of complex functionally graded material (FGM) sectioned beams. Portions of these beams may be made from different materials with possibly different gradients of material properties. The present work proposes models to investigate the free vibration of FGM sectioned beams based on onedimensional (1D) finite element analysis. For this purpose, a sample beam is divided into discrete elements, and the total energy stored in each element during vibration is computed by considering either the Timoshenko or Euler-Bernoulli beam theory. Then, Hamilton's principle is used to derive the equations of motion for the beam. The effects of material properties and dimensions of FGM sections on the beam's natural frequencies and their corresponding mode shapes are then investigated based on a dynamic Timoshenko model (TM). The presented model is validated by comparison with three-dimensional (3D) finite element simulations of the first three mode shapes of the beam.     

Winkler地基模型上功能梯度材料涂层的摩擦接触问题

研究Winker地基模型上功能梯度材料涂层在一刚性圆柱形冲头作用下的摩擦接触问题。功能梯度材料涂层表面作用有法线向和切线向集中作用力。假设材料非均匀参数呈指数形式变化,泊松比为常量,利用Fourier积分变换技术将求解模型的接触问题转化为奇异积分方程组,再利用切比雪夫多项式对所得奇异积分方程组进行数值求解。最后,给出了功能梯度材料非均匀参数、摩擦系数、Winker地基模型刚度系数及冲头曲率半径对接触应力分布和接触区宽度的影响情况。     

粘贴压电层功能梯度材料Timoshenko梁的热过屈曲分析

研究了上下表面粘贴压电层的功能梯度材料Timoshenko梁在升温及电场作用下的过屈曲行为。在精确考虑轴线伸长和一阶横向剪切变形的基础上,建立了压电功能梯度Timoshenko层合梁在热-电-机械载荷作用下的几何非线性控制方程。其中,假设功能梯度的材料性质沿厚度方向按照幂函数连续变化,压电层为各向同性均匀材料。采用打靶法数值求解所得强非线性边值问题,获得了在均匀电场和横向非均匀升温场内两端固定Timoshenko梁的静态非线性屈曲和过屈曲数值解。并给出了梁的变形随热、电载荷及材料梯度参数变化的特性曲线。结果表明,通过施加电压在压电层产生拉应力可以有效地提高梁的热屈曲临界载荷,延缓热过屈曲发生。由于材料在横向的非均匀性,即使在均匀升温和均匀电场作用下,也会产生拉-弯耦合效应。但是对于两端固定的压电-功能梯度材料梁,在横向非均匀升温下过屈曲变形仍然是分叉形的。     

弹性地基加肋功能梯度板自由振动分析的无网格法

基于一阶剪切变形理论和移动最小二乘近似研究Winkler弹性地基上加肋功能梯度板的固有频率。假设功能梯度板的材料性质沿厚度方向按幂函数连续变化,基于物理中面和移动最小二乘近似分别推导功能梯度板和肋条的动能和势能,再通过引入位移协调条件,建立板和肋条节点参数转换关系,叠加两者的总能量,然后利用Hamilton原理推导加肋功能梯度板自由振动控制方程。采用完全转换法施加边界条件。通过将本文的计算结果与有限元以及文献的结果对比,验证方法的收敛性以及准确性。     

Numerical study on wave propagation in a low-rigidity elastic medium considering the effects of gravity

We discuss the effects of vertical gravity force on wave propagation when a material is intermediate between solid and fluid, especially we focus on what kinds of phase are generated and how it propagates on the surface. We introduce gravity terms into the 2D linear finite element method in order to account for the contribution from the gravity. Numerical simulations are conducted for a half-space model and a two-layered, single horizontal layer overlain on a half-space, model. Both models are compared between the results including and excluding the viscosity. The fastest phase propagating from a surface point source, a leaking Rayleigh wave for usual elastic material, is transformed into an interesting phase including some common features to the gravity wave when the gravity effect becomes significant. The viscosity does not affect the fastest phases, whereas it affects other latter phases appearing only for the two-layered model.     

弹性地基上含孔隙功能梯度材料板的自由振动和动力响应

为研究弹性地基上含孔隙的材料特性沿厚度呈Sigmoid函数变化的功能梯度材料(S-FGM)板的振动特性,本文基于改进的Voigt模型,分别建立了孔隙为均匀分布和非均匀分布两种类型的功能梯度材料的物性参数模型.根据复合材料薄板理论导出了弹性地基上含孔隙的功能梯度材料板的运动方程,用伽辽金法寻求四边简支边界条件下板自由振动...     

Unified two-phase nonlocal formulation for vibration of functionally graded beams resting on nonlocal viscoelastic Winkler-Pasternak foundation

A nonlocal study of the vibration responses of functionally graded (FG) beams supported by a viscoelastic Winkler-Pasternak foundation is presented. The damping responses of both the Winkler and Pasternak layers of the foundation are considered in the formulation, which were not considered in most literature on this subject, and the bending deformation of the beams and the elastic and damping responses of the foundation as nonlocal by uniting the equivalently differential formulation of well-posed strain-driven (ε-D) and stress-driven (σ-D) two-phase local/nonlocal integral models with constitutive constraints are comprehensively considered, which can address both the stiffness softening and toughing effects due to scale reduction. The generalized differential quadrature method (GDQM) is used to solve the complex eigenvalue problem. After verifying the solution procedure, a series of benchmark results for the vibration frequency of different bounded FG beams supported by the foundation are obtained. Subsequently, the effects of the nonlocality of the foundation on the undamped/damping vibration frequency of the beams are examined.     

Free vibration of functionally graded beams based on both classical and first-order shear deformation beam theories

The free vibration of functionally graded material （FGM） beams is studied based on both the classical and the first-order shear deformation beam theories. The equations of motion for the FGM beams are derived by considering the shear deforma- tion and the axial, transversal, rotational, and axial-rotational coupling inertia forces on the assumption that the material properties vary arbitrarily in the thickness direction. By using the numerical shooting method to solve the eigenvalue problem of the coupled ordinary differential equations with different boundary conditions, the natural frequen- cies of the FGM Timoshenko beams are obtained numerically. In a special case of the classical beam theory, a proportional transformation between the natural frequencies of the FGM and the reference homogenous beams is obtained by using the mathematical similarity between the mathematical formulations. This formula provides a simple and useful approach to evaluate the natural frequencies of the FGM beams without dealing with the tension-bending coupling problem. Approximately, this analogous transition can also be extended to predict the frequencies of the FGM Timoshenko beams. The numerical results obtained by the shooting method and those obtained by the analogous transformation are presented to show the effects of the material gradient, the slenderness ratio, and the boundary conditions on the natural frequencies in detail.     

Buckling analysis of functionally graded nanobeams under non-uniform temperature using stress-driven nonlocal elasticity

In this work, the size-dependent buckling of functionally graded(FG)Bernoulli-Euler beams under non-uniform temperature is analyzed based on the stressdriven nonlocal elasticity and nonlocal heat conduction. By utilizing the variational principle of virtual work, the governing equations and the associated standard boundary conditions are systematically extracted, and the thermal effect, equivalent to the induced thermal load, is explicitly assessed by using the nonlocal heat conduction law. The ...     

Surface and thermal effects on vibration of embedded alumina nanobeams based on novel Timoshenko beam model

This paper deals with the free vibration analysis of circular alumina （Al2O3） nanobeams in the presence of surface and thermal effects resting on a Pasternak foun- dation. The system of motion equations is derived using Hamilton＇s principle under the assumptions of the classical Timoshenko beam theory. The effects of the transverse shear deformation and rotary inertia are also considered within the framework of the mentioned theory. The separation of variables approach is employed to discretize the governing equa- tions which are then solved by an analytical method to obtain the natural frequencies of the alumina nanobeams. The results show that the surface effects lead to an increase in the natural frequency of nanobeams as compared with the classical Timoshenko beam model. In addition, for nanobeams with large diameters, the surface effects may increase the natural frequencies by increasing the thermal effects. Moreover, with regard to the Pasternak elastic foundation, the natural frequencies are increased slightly. The results of the present model are compared with the literature, showing that the present model can capture correctly the surface effects in thermal vibration of nanobeams.     

Wave pattern characteristics of a two-phase nozzle flow by shock propagation

In this paper, the wave pattern characteristics of shock-induced two-phase nozzle flows with the quiescent or moving dusty gas ahead of the incident-shock front has been investigated by using high-resolution numerical method. As compared with the corresponding results in single-phase nozzle flows of the pure gas, obvious differences between these two kinds of flows can be obtained. Received 14 June 1996 / Accepted 19 October 1996     

Flapwise bending vibration analysis of a rotating double-tapered Timoshenko beam

In this study, free vibration analysis of a rotating, double-tapered Timoshenko beam that undergoes flapwise bending vibration is performed. At the beginning of the study, the kinetic- and potential energy expressions of this beam model are derived using several explanatory tables and figures. In the following section, Hamilton’s principle is applied to the derived energy expressions to obtain the governing differential equations of motion and the boundary conditions. The parameters for the hub radius, rotational speed, shear deformation, slenderness ratio, and taper ratios are incorporated into the equations of motion. In the solution, an efficient mathematical technique, called the differential transform method (DTM), is used to solve the governing differential equations of motion. Using the computer package Mathematica the effects of the incorporated parameters on the natural frequencies are investigated and the results are tabulated in several tables and graphics.     

Modal analysis of cracked continuous Timoshenko beam made of functionally graded material

Abstract

The article addresses development of the Transfer Matrix Method (TMM) for free vibration of cracked continuous Timoshenko beam made of Functionally Graded Material (FGM). The governing equations of free vibration are established for the beam based on the power law of material grading, actual position of neutral plane and double spring model of crack. There is conducted frequency equation of the beam with intermediate rigid supports using the TMM after the transverse displacements at rigid supports have been disregarded. Therefore, the frequency equation is simplified and becomes more useful to compute natural frequencies of continuous FGM Timoshenko beam with a number of cracks. The obtained numerical results show the essential effect of cracks, material properties and also number of spans on natural frequencies of the beam.     

Free vibration analysis of functionally graded material beams based on Levinson beam theory

Free vibration response of functionally graded material (FGM) beams is studied based on the Levinson beam theory (LBT). Equations of motion of an FGM beam are derived by directly integrating the stress-form equations of elasticity along the beam depth with the inertial resultant forces related to the included coupling and higherorder shear strain. Assuming harmonic response, governing equations of the free vibration of the FGM beam are reduced to a standard system of second-order ordinary differential equations associated with boundary conditions in terms of shape functions related to axial and transverse displacements and the rotational angle. By a shooting method to solve the two-point boundary value problem of the three coupled ordinary differential equations, free vibration response of thick FGM beams is obtained numerically. Particularly, for a beam with simply supported edges, the natural frequency of an FGM Levinson beam is analytically derived in terms of the natural frequency of a corresponding homogenous Euler-Bernoulli beam. As the material properties are assumed to vary through the depth according to the power-law functions, the numerical results of frequencies are presented to examine the effects of the material gradient parameter, the length-to-depth ratio, and the boundary conditions on the vibration response.