Computational approach of piezoelectric shells by the GDQ method

A piezoelectric laminated cylindrical shell with shear rotations effect under the electromechanical loads and four sides simply supported boundary condition was studied by using the two-dimensional generalized differential quadrature (GDQ) computational method. The typical hybrid composite shells with 3-layered cross-ply [90°/0°/90°] graphite–epoxy laminate and bounded PVDF layers are considered under the sinusoidal pressure loads and electric potentials on the shell. The governing partial differential equation with first-order shear deformation theory in terms of mid-surface displacements and shear rotations can be expressed in series equations by the GDQ formulation. Thus we obtain the GDQ numerical solutions of non-dimensional displacement and stresses at center position of laminated piezoelectric shells. Displacement is generally affected by the thickness of laminated piezoelectric shells under the action of mechanical load. Stresses are generally affected by the thickness and the length of laminated piezoelectric shells under the actions of mechanical load and electric potential.     

Dynamic instability response of smart composite material

Laminated composite is common to replace traditional metals in today's industries due to its high specific strength. Shape memory alloy has been used to improve structural behaviours such as buckling, vibration and fatigue through its well‐ known property of shape memory effect. This ability of shape memory alloy to improve the parametric instability behaviour of laminated composite plate has not been studied in the past and as such, this study is conducted. Here, shape memory alloy wires are embedded within the outer layers of the laminated composite plates. The Mathieu‐Hill equation for the parametric instability of the shape memory alloy composite plate has been developed using finite element method based on the first order shear deformation theory. The formulation is validated and parametric studies have been conducted to investigate the effect of shape memory alloy on the dynamic instability behaviour of the composite that corresponds to factors such as static load factor, thickness of the plate and boundary conditions. The study shows that shape memory alloy improves significantly the dynamic instability behaviour of the laminated composite plate by shifting the instability chart to the right. The effect of shape memory alloy can increase the frequency centre of the instability chart by more than 100 %.     

不同边界条件下转动锥壳的参激失稳特性分析

采用Haar小波方法结合Floquet指数法对不同边界条件下转动锥壳的参激振动稳定性进行了分析。基于Love一阶近似壳体理论,给出了周期性载荷作用下转动锥壳的动力学控制微分方程,采用Haar小波离散方法将其转化为具有周期性时变系数的Mathieu-Hill型常微分方程组。考虑到Bolotin法不能应用于陀螺系统的参激失稳特性分析,以及多尺度法受限于小参数情形的事实,该研究采用了对参激系统普遍适用的Floquet指数法对转动锥壳的参激振动稳定性进行分析。通过与其他文献结果的对比,验证了所采用模型及稳定性分析方法的正确性。在此基础上,讨论了固支-固支、简支-简支、固支-简支和简支-固支等几种不同边界条件下转速和半顶角对转动锥壳不稳定区的影响。     

Laminated composite rectangular and annular plates: A GDQ solution for static analysis with a posteriori shear and normal stress recovery

The Generalized Differential Quadrature (GDQ) Method is applied to study laminated composite degenerate shell panels such as rectangular and annular plates. The theoretical treatment is maintained general in order to expose in a unique way the procedure adopted to obtain the stress profiles through the thickness of plates without specifying the equations for rectangular and annular plates. By simply imposing some geometrical relations the equations governing the problem of plates under consideration, that are degenerate shells, are inferred from the theory of shells of revolution. The mechanical model is based on the so called First-order Shear Deformation Theory (FSDT) deduced from the three-dimensional theory in order to analyse the above moderately thick structural elements. The solution is given in terms of generalized displacement components of points lying on the middle surface of the plate. After the solution of the fundamental system of equations in terms of displacements and rotations, the generalized strains and stress resultants are evaluated by applying the Differential Quadrature rule to the generalized displacements. The transverse shear and normal stress profiles through the laminate thickness are reconstructed a posteriori by using local three-dimensional elasticity equilibrium equations. No preliminary recovery or regularization procedure on the extensional and flexural strain fields is needed when the Differential Quadrature technique is used. By using GDQ procedure through the thickness, the reconstruction procedure needs only to be corrected to properly account for the boundary equilibrium conditions. In order to verify the accuracy of the present method, GDQ results are compared with the ones obtained with semi-analytical formulations and with 3D finite element methods. Stresses of several composite plates are evaluated. Very good agreement is observed without using mixed formulations and higher order kinematical models. Various examples of stress profiles for rectangular and annular plate elements are presented to illustrate the validity and the accuracy of GDQ method.     

General higher-order layer-wise theory for free vibrations of doubly-curved laminated composite shells and panels

ABSTRACT

The present article illustrates a general formulation for a higher-order layer-wise theory related to the analysis of the free vibrations of thick doubly-curved laminated composite shells and panels. The theoretical framework relates to the dynamic analysis of shell structures by using a general displacement field based on the Carrera Unified Formulation (CUF), including the stretching effect for each layer. The order of the expansion along the thickness direction is taken as a free parameter. The starting point of the present general higher-order layer-wise formulation is to propose a kinematic assumption, with an arbitrary number of degrees of freedom. The main aim of this work is to determine the explicit fundamental operators that can be used for the layer-wise (LW) approach. These fundamental operators are obtained for the first time by the author and are related to motion equations of doubly-curved shells described in an orthogonal curvilinear co-ordinate system. The free vibration shell and panel problems are computationally solved using the generalized differential quadrature (GDQ) and generalized integral quadrature (GIQ) techniques. The numerical results are compared with recent papers in the literature and commercial finite element codes.     

Rapid heating of FGM rectangular plates

Thermally induced vibrations of functionally graded material rectangular plates are investigatedin this research. The thermomechanical properties of the plate are assumed to be temperature and positiondependent. Dependency on temperature is expressed based on theTouloukian formula, and position dependencyis written as a power-law function. The ceramic-rich surface of the plate is subjected to temperature rise orheat flux, whereas the metal rich surface is kept at reference temperature or thermally insulated. Temporalevolution of the temperature profile across the plate thickness is obtained by the solution of one-dimensionalheat conduction equation. This equation is originally nonlinear since temperature dependency of thermalconductivity is taken into account. The solution of this equation is obtained by means of the generalizeddifferential quadrature (GDQ) accompanied with the successive Runge–Kutta algorithm in time domain. Themotion equations of the plate are obtained based on the first-order shear deformation theory of plates under smallstrains and small deformations assumptions. Hamilton’s principle is used to establish the motion equations.These equations are discreted in the plate domain bymeans of the two-dimensional GDQ method. The resultingequations are linear time-dependent coupled equations which are traced in time by means of the Newmarktime-marching method. Conducting comparison studies to assure the validity and accuracy of the proposedmodel, parametric studies are carried out to examine the influences of temperature dependency, thermal andmechanical boundary conditions, power-law index, plate geometry and boundary conditions. It is shown thatthermally induced vibrations exist for thin plates.     

Nonlinear dynamic stability of laminated composite shells integrated with piezoelectric layers in thermal environment

This paper reports the nonlinear dynamic stability characteristics of laminated composite cylindrical (CYL) and spherical (SPH) shells integrated with piezoelectric layers using the finite element method. The shells are subjected to a thermal environment in addition to the in-plane periodic load and the electric load. The theoretical formulation considers Sanders?? approximation for doubly curved shells, and von Kármán type nonlinear strains are incorporated into the first-order shear deformation theory (FSDT). The formulation includes the effects of transverse shear, in-plane and rotatory inertia. The in-plane periodic load is taken as the parametric excitation in the governing equation. The nonlinear matrix amplitude equation is obtained by employing Galerkin??s method. The correctness of the formulation is established by comparing the authors?? results with those available in the published literature. Detailed parametric studies are carried out to investigate the effects of different parameters on the dynamic stability characteristics of laminated composite shells.     

薄膜结构气弹动力稳定性研究

将扁壳的无矩理论和流体的理想势流理论结合起来对薄膜结构的气弹动力稳定性进行了研究,提出了结构失稳的判别准则,确定了结构失稳临界风速。首先应用扁壳的无矩理论建立了薄膜结构的动力平衡方程。然后假设来流为均匀的理想势流,考虑流固耦合作用,对风向沿结构拱向和垂向时分别采用不同的气弹模型确定了作用于薄膜表面的气动力,得到了两种情况下薄膜结构的气弹动力耦合作用方程。利用Bubnov-Galerkin方法将此耦合作用方程转化为一常系数二阶微分方程,并根据Routh-Hurwitz稳定性准则确定了薄膜的失稳临界风速。最后通过对临界风速的影响因素进行分析,得到了一些重要结论,并提出了防止薄膜结构气弹失稳的一些基本措施。     

Free vibration analysis of rotating functionally graded cylindrical shells in thermal environment

The free vibration analysis of rotating functionally graded (FG) cylindrical shells subjected to thermal environment is investigated based on the first order shear deformation theory (FSDT) of shells. The formulation includes the centrifugal and Coriolis forces due to rotation of the shell. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The initial thermo-mechanical stresses are obtained by solving the thermoelastic equilibrium equations. The equations of motion and the related boundary conditions are derived using Hamilton’s principle. The differential quadrature method (DQM) as an efficient and accurate numerical tool is adopted to discretize the thermoelastic equilibrium equations and the equations of motion. The convergence behavior of the method is demonstrated and comparison studies with the available solutions in the literature are performed. Finally, the effects of angular velocity, Coriolis acceleration, temperature dependence of material properties, material property graded index and geometrical parameters on the frequency parameters of the FG cylindrical shells with different boundary conditions are investigated.     

Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations

The Generalized Differential Quadrature (GDQ) method is applied to study the dynamic behavior of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature from the beginning of the theory formulation a generalization of the kinematical model is adopted for the Reissner–Mindlin and Toorani–Lakis theory. By so doing a generalization of the theory of anisotropic doubly-curved shells and panels of revolution is proposed. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. The Differential Quadrature (DQ) rule is introduced to determine the geometric parameters of the structures with a free-form meridian. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the general formulation and the Classical Reissner–Mindlin and Classical Toorani–Lakis theory are presented. New results are presented in order to investigate the effects of the Winkler modulus, the Pasternak modulus and the inertia of the elastic foundation on the free vibrations of anisotropic shells of revolution with a free-form meridian.     

Dynamic stability of stiffened laminated composite plates and shells subjected to in-plane pulsating forces

The dynamic stability of laminated composite stiffened or non-stiffened plates and shells due to periodic in-plane forces at boundaries is investigated in this paper. A three-dimensional (3-D) degenerated shell element and a 3-D degenerated curved beam element are used to model plates/shells and stiffeners, respectively. The characteristic equations to find the natural frequencies, buckling loads and their corresponding mode shapes are obtained from the finite element equation of motion. Then, the method of Hill's infinite determinants or the method of multiple scales is applied to analyse the dynamic instability regions. Numerical results are presented to demonstrate the effects of various parameters, such as skew angle, lamination scheme, stiffened scheme, in-plane force type and curvature of cylindrical shell, on the dynamic stability of stiffened and non-stiffened plates and shells subjected to in-plane pulsating forces at boundaries.     

2-D GDQ solution for free vibrations of anisotropic doubly-curved shells and panels of revolution

In this paper, the Generalized Differential Quadrature (GDQ) method is applied to study the dynamic behaviour of laminated composite doubly-curved shells of revolution. The First-order Shear Deformation Theory (FSDT) is used to analyse the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature a generalization of the Reissner–Mindlin theory, proposed by Toorani and Lakis, is adopted. The governing equations of motion, written in terms of stress resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. The discretization of the system by means of the Differential Quadrature (DQ) technique leads to a standard linear eigenvalue problem, where two independent variables are involved. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the Reissner–Mindlin and Toorani–Lakis theory are presented. Furthermore, GDQ results are compared with those presented in literature and the ones obtained by using commercial programs such as Abaqus, Ansys, Nastran, Straus and Pro/Mechanica. Very good agreement is observed.     

波纹膜片的非线性稳定

应用轴对称旋转扁壳的基本方程,研究了在任意载荷作用下各种边界条件的波纹膜片的非线性稳定问题。采用格林函数方法,将扁壳的非线性微分方程组化为非线性积分方程组。再使用展开法求出格林函数,即将格林函数展开成特征函数的级数形式,积分方程就成为具有退化核的形式,从而容易得到非线性代数方程组。应用牛顿法求解非线性代数方程组时,为了保证迭代的收敛性,选取位移作为控制参数,逐步增加位移,求得相应的载荷。作为算例,首先研究了带中心平台三波纹膜片的局部失稳现象,然后讨论了由于缺陷的存在,波纹膜片有可能出现的极值点失稳,这是一种类似扁球壳的总体失稳现象。解答可供波纹膜片的设计参考。     

Thin shells with finite rotations formulated in biot stresses: Theory and finite element formulation

A bending theory for thin shells undergoing finite rotations is presented, and its associated finite element model is described. The kinematic assumption is based on a shear elastic Reissner-Mindlin theory. The starting point for the derivation of the strain measures are the resultant equilibrium equations and the associated principle of virtual work. Within this formulation the polar decomposition of the shell material deformation gradient leads to symmetric strain measures. The associated work-conjugate stress resultants and stress couples are integrals of the Biot stress tensor. This tensor is invariant with respect to rigid body motions and, therefore, appropriate for the formulation of constitutive equations. Finite rotations are introduced via Eulerian angles. The finite element discretization of arbitrary shells is based on the isoparametric concept formulated with respect to a plane reference configuration. The numerical model is applied to different non-linear plate and shell problems and compared with existing formulations. Due to a consistent linearization, the step size of a load increment is only limited by the local convergence behaviour of Newton's method.     

Numerical investigation of functionally graded cylindrical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery

A 2D Unconstrained Third Order Shear Deformation Theory (UTSDT) is presented for the evaluation of tangential and normal stresses in moderately thick functionally graded cylindrical shells subjected to mechanical loadings. Eight types of graded materials are investigated. The functionally graded material consists of ceramic and metallic constituents. A four parameter power law function is used. The UTSDT allows the presence of a finite transverse shear stress at the top and bottom surfaces of the graded cylindrical shell. In addition, the initial curvature effect included in the formulation leads to the generalization of the present theory (GUTSDT). The Generalized Differential Quadrature (GDQ) method is used to discretize the derivatives in the governing equations, the external boundary conditions and the compatibility conditions. Transverse and normal stresses are also calculated by integrating the three dimensional equations of equilibrium in the thickness direction. In this way, the six components of the stress tensor at a point of the cylindrical shell or panel can be given. The initial curvature effect and the role of the power law functions are shown for a wide range of functionally cylindrical shells under various loading and boundary conditions. Finally, numerical examples of the available literature are worked out.     

Dynamic stability of linear parametrically excited twisted Timoshenko beams under periodic axial loads

In this paper, the parametric instability of twisted Timoshenko beams with various end conditions and under an axial pulsating force is studied. The equations of motion in the twisted frame are derived using a finite element method. Based on Bolotin??s method, a set of second-order ordinary differential equations with periodic coefficients of Mathieu?CHill type is formed to determine the instability regions for twisted Timoshenko beams. A dynamic instability index is defined and used as an instability measure to study the influence of various parameters. The effects of beam length, inertia ratio, pre-twist angle, dynamic component of axial force and restraint condition on the instability regions and dynamic instability index of the twisted beam are investigated and discussed.     

Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations

The Generalized Differential Quadrature (GDQ) method is applied to study the dynamic behavior of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature from the beginning of the theory formulation a generalization of the kinematical model is adopted for the Reissner–Mindlin and Toorani–Lakis theory. By so doing a generalization of the theory of anisotropic doubly-curved shells and panels of revolution is proposed. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. The Differential Quadrature (DQ) rule is introduced to determine the geometric parameters of the structures with a free-form meridian. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the general formulation and the Classical Reissner–Mindlin and Classical Toorani–Lakis theory are presented. New results are presented in order to investigate the effects of the Winkler modulus, the Pasternak modulus and the inertia of the elastic foundation on the free vibrations of anisotropic shells of revolution with a free-form meridian.     

Interlaminar stress analysis in general thick composite cylinder subjected to nonuniform distributed radial pressure

Interlaminar stresses in thick a composite cylinder with general layer stacking subjected to uniform and nonuniform distributed radial pressure are studied. The layerwise theory of Reddy is employed for formulation of the problem. An analytical method is presented for solving the governing equations. To increase accuracy, interlaminar stresses are obtained by integrating the equilibrium equation of elasticity. After a convergence study, the accuracy of the layerwise laminate theory is investigated using the predictions of finite element method. Predictions of Hooke's law and integration method for interlaminar stresses are compared. Uniform and nonuniform internal and external loads are considered and a parametric study is done for various cylinders.     

Pulsating fluid induced dynamic instability of visco-double-walled carbon nano-tubes based on sinusoidal strain gradient theory using DQM and Bolotin method

This research deals with the dynamic instability analysis of double-walled carbon nanotubes (DWCNTs) conveying pulsating fluid under 2D magnetic fields based on the sinusoidal shear deformation beam theory (SSDBT). In order to present a realistic model, the material properties of DWCNTs are assumed viscoelastic using Kelvin–Voigt model. Considering the strain gradient theory for small scale effects, a new formulation of the SSDBT is developed through the Gurtin–Murdoch elasticity theory in which the effects of surface stress are incorporated. The surrounding elastic medium is described by a visco-Pasternak foundation model, which accounts for normal, transverse shear and damping loads. The van der Waals interactions between the adjacent walls of the nanotubes are taken into account. The size dependent motion equations and corresponding boundary conditions are derived based on the Hamilton’s principle. The differential quadrature method in conjunction with Bolotin method is applied for obtaining the dynamic instability region. The detailed parametric study is conducted, focusing on the combined effects of the nonlocal parameter, magnetic field, visco-Pasternak foundation, Knudsen number, surface stress and fluid velocity on the dynamic instability of DWCNTs. The results depict that the surface stress effects on the dynamic instability of visco-DWCNTs are very significant. Numerical results of the present study are compared with available exact solutions in the literature. The results presented in this paper would be helpful in design and manufacturing of nano/micro mechanical systems in advanced biomechanics applications with magnetic field as a parametric controller.