Axisymmetric response of circular plates with piezoelectric layers: an exact solution

Electromechanical responses of symmetric circular laminates consisting of piezoelectric layers are studied, and the influence of surface and interlayer electrodes are involved. The laminates are traction-free on the top and bottom surfaces, but may be subjected to external forces at the lateral edge and to voltages applied across certain layers. Under axisymmetric deformation conditions, an approximate model which employs Kirchhoff hypothesis and incorporates the charge equation of electrostatic is established. Then, a closed-form three-dimensional solution of the laminates is generated in a very straightforward manner by the solution of the approximate model. The three-dimensional solution fulfills all field equations and interface or surface conditions as well as the specified electric edge boundary conditions; the only restriction is that the mechanical edge boundary conditions are satisfied in an average manner, rather than point by point. Thus, according to Saint-Venant's principle the proposed solution is exact in the interior region of the laminates.     

Three-dimensional analysis of thermal stresses in four-parameter continuous grading fiber reinforced cylindrical panels

In this paper, three-dimensional analysis of thermal stresses in four-parameter continuous grading fiber reinforced cylindrical panel subjected to thermal load is studied. The cylindrical panel is assumed to be made of an orthotropic material. The continuous grading fiber reinforced panel has a smooth variation in matrix volume fraction in the radial direction. A generalization of the power-law distribution presented in literature is proposed. Symmetric and asymmetric volume fraction profiles are presented in this paper. Suitable temperature and displacement functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the thermoelastic equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which can be solved by generalized differential quadrature method. The fast rate of convergence of the method is demonstrated and to validate the results, comparisons are made with the available solutions for orthotropic shells. The main contribution of this work is to illustrate the influence of the power-law exponent, of the power-law distribution choice and of the choice of the four parameters on the thermal behaviour of continuous grading fiber reinforced cylindrical panels.     

Critical conditions for thermal explosion in a plate with a nonlinear heat source

The exact analytical solution of the nonstationary heat conduction problem with a nonlinear internal heat source at the nonsymmetric third-type boundary conditions was obtained by the method of variable separation. The relations between the boundary condition parameters and a heat source separating the stationary processes from processes of an unlimited temperature increase (thermal explosion) were found. For known physical properties of medium, the maximum allowable value of the specific power of a heat source has been found. If nonsymmetric boundary conditions are exceeded on wall surfaces, it can lead to thermal destruction.     

Stress in piezoelectric hollow sphere with thermal gradient

The piezoelectric phenomenon has been exploited in science and engineering for decades. Recent advances in smart structures technology have led to a resurgence of interest in piezoelectricity, and in particular, in the solution of fundamental boundary value problems. In this paper, we develop an analytic solution to the axisymmetric problem of a radially polarized, spherically isotropic piezoelectric hollow sphere. The sphere is subjected to uniform internal pressure, or uniform external pressure, or both and thermal gradient. There is a constant thermal difference between its inner and outer surfaces. An analytic solution to the governing equilibrium equations (a coupled system of second-order ordinary differential equations) is obtained. On application of the boundary conditions, the problem is reduced to solving a system of linear algebraic equations. Finally, the stress distributions in the sphere are obtained numerically for two piezoceramics.     

A simple procedure for determining spatial and transient variations of cooling rate within a specimen during cryopreservation. Part 1: Analysis

The cryopreservation of biological materials may involve cooling continuously in the liquid state (as in vitrification) or nucleation of ice, followed by thermal equilibration at a selected sub-zero temperature and subsequent cooling in the solid phase. Both of these processes may be approximated by a single-phase heat transport analysis, the most critical parameter of which, for biological survival, is the cooling rate. The Fourier series solution for the cooling rate in a specimen is presented for single-phase heat transfer in cartesian, cylindrical and spherical coordinates; extensive tables of the series constants and of the roots of the transcendental equations have been computed that include the cases of most relevance to cryopreservation. In particular, data are provided to calculate the cooling rate at process states close to the initial conditions, that is at small Fourier numbers. Graphical results of the analysis are given in Part 2.     

Free vibration analysis of solar functionally graded plates with temperature-dependent material properties using second order shear deformation theory

Second-order shear deformation theory (SSDT) is employed to analyze vibration of temperature-dependent solar functionally graded plates (SFGP’s). Power law material properties and linear steady-state thermal loads are assumed to be graded along the thickness. Two different types of SFGP’s such as ZrO₂/Ti-6Al-4V and Si₃N₄/SUS304 are considered. Uniform, linear, nonlinear, heat-flux and sinusoidal thermal conditions are imposed at the upper and lower surface for simply supported SFGPs. The energy method is applied to derive equilibrium equations, and solution is based on Fourier series that satisfy the boundary conditions (Navier’s method). Non-dimensional results are compared for temperature-dependent and temperature-independent SFGP’s and validated with known results in the literature. Numerical results indicate the effect of material composition, plate geometry, and temperature fields on the vibration characteristics and mode shapes. The results obtained using the SSDT are very close to results from other shear deformation theories.     

Free vibration of transversely isotropic piezoelectric circular cylindrical panels

This paper presents an exact three-dimensional free vibration analysis of a transversely isotropic piezoelectric circular cylindrical panel. The general solution for coupled equations for piezoelectric media that was recently proposed by Ding et al. (Int. J. Solids Struct. 33 (1996) 2283) is employed. By using the variable separation method, three-dimensional exact solutions are obtained under several boundary conditions. Numerical results are finally presented and compared with available data in literature. The results show the non-dimensional frequencies of the piezoelectric panel are bigger than that of the non-piezoelectric one.     

Radial Lip Seal Analysis—Finite Element Solution for a Body of Revolution

To deal with the complex geometry of radial lip seals, in particular the metal-to-rubber boundary and multiple lips, the elasticity equations for a body of revolution are an important step to improve the accuracy of the predicted solution. A nonlinear strain displacement finite element solution of the elasticity equations for a body of revolution is presented here. Following Zienkiewia. (1), the equations are derived from the work equivalence principle generalized to nonlinear problems. Realistic levels of Poisson's ratio present no difficulty when a technique of reduced integration is used (2), The elastic law is Hooke's law with thermal expansion.

External forces can be volume forces or surface forces. Both are converted into consistent nodal forces. Boundary conditions can be any combination of forces or displacements on any given node. This paper presents the equations and the solution by the finite element method as well as an extensive verification of the program prediction for thermal shrinkage of the seal from mold to ambient temperature, showing remarkable agreement with experiments in a large number of cases. Comparison of the prediction of the program for radial load and axial movement of the lip against experiment are also given, as well as comparison with the results of the previous code for conical shells.     

Buckling of rectangular plates with various boundary conditions loaded by non-uniform inplane loads

In the present paper, buckling loads of rectangular composite plates having nine sets of different boundary conditions and subjected to non-uniform inplane loading are presented considering higher order shear deformation theory (HSDT). As the applied inplane load is non-uniform, the buckling load is evaluated in two steps. In the first step the plane elasticity problem is solved to evaluate the stress distribution within the prebuckling range. Using the above stress distribution the plate buckling equations are derived from the principle of minimum total potential energy. Adopting Galerkin's approximation, the governing partial differential equations are converted into a set of homogeneous linear algebraic equations. The critical buckling load is obtained from the solution of the associated linear eigenvalue problem. The present buckling loads are compared with the published results wherever available. The buckling loads obtained from the present method for plate with various boundary conditions and subjected to non-uniform inplane loading are found to be in excellent agreement with those obtained from commercial software ANSYS. Buckling mode shapes of plate for different boundary conditions with non-uniform inplane loadings are also presented.     

Analytical solution of non-Fourier heat conduction problem on a fin under periodic boundary conditions

Fourier and hyperbolic models of heat transfer on a fin that is subjected to a periodic boundary condition are solved analytically. The differential equation in Fourier and non-Fourier models is solved by the Laplace transform method. The temperature distribution on the fin is obtained using the residual theorem in a complex plan for the inverse Laplace transform method. The thermal shock is generated at the base of the fin, which moves toward the tip of the fin and is reflected from the tip. The current study of various parameters on the thermal shock location shows that relaxation time has a great influence on the temperature distribution on the fin. An unsteady boundary condition in the base fin caused the shock, which is generated continuously from the base and has interacted with the other reflected thermal shocks. Results of the current study show that the hyperbolic heat conduction equation can violate the second thermodynamic law under some unsteady boundary conditions.     

Flow and thermal characteristics of a circular porous slider bearing

An incompressible newtonian fluid is forced through the porous bottom of a circular slider which is moving laterally on a horizontal plane. The effects of mass injection and lateral velocity on the heat generated by viscous dissipation are investigated by solving the governing boundary layer equations using numerical integration techniques. Numerical results are presented for the velocity and temperature distributions as well as for the wall derivatives of the velocity and thermal functions. The range of Prandtl numbers investigated varies from 0.7 to 10 while the cross-flow Reynolds numbers and translational Reynolds numbers range from 0.01 to 50 and 1 to 1000 respectively.     

Three-Dimensional Thermohydrodynamic Analyses of Rayleigh Step Air Foil Thrust Bearing with Radially Arranged Bump Foils

A three-dimensional (3D) thermohydrodynamic (THD) model for air foil thrust bearings (AFTBs) is presented. The nonisothermal Reynolds equation is solved using pressure boundary conditions at the cooling air plenum considering local temperature-dependent viscosity and density. Air film temperature is calculated using the 3D energy equation with thermal boundary conditions at the top foil, thrust runner, and top foil’s leading edge. The cooling air plenum distributes the cooling air to multiple radially arranged cooling channels. The plenum temperature and pressure are found from mass and energy balance equations applied to the plenum. Temperature fields of the top foil, bump foils, thrust disc runner, bearing plate, and cooling air channels are also solved through appropriate energy balance equations with their surroundings. A robust computational algorithm with multiple iteration loops was developed to find all the temperature fields. THD analyses were performed for AFTB with outer radius of 50 mm up to 100,000 rpm. As the cooling air source pressure is increased, the plenum pressure also increases and its temperature decreases due to more cooling capacity. However, cooling effectiveness is not necessarily proportional to the pressure because the flow residence time inside the cooling channels is inversely proportional to the pressure. The analyses show that the thrust disc temperature is a parabolic function with speed, and thermal expansions of the thrust disc and thrust plates contribute to the most significant driving force of thermal instability. Optimum cooling air pressure was found around 12,500 Pa for the proposed AFTB design at the reference simulation condition.     

Boundary and thermal non-equilibrium effects on convective instability in an anisotropic porous layer

The effects of boundary and local thermal non-equilibrium on the criterion for the onset of convection in a sparsely packed horizontal anisotropic porous layer are investigated. A two-field temperature model each representing the solid and fluid phases separately is used and the flow in the porous medium is described by the Brinkman extended-Darcy model. The lower boundary is rigid, while the upper boundary is considered to be either rigid or free with fixed temperature conditions at the boundaries. The stability equations are solved numerically using the Galerkin method to extract the critical stability parameters. The influence of local thermal non-equilibrium, mechanical and thermal anisotropy parameters representing the fluid and solid phases is assessed on the stability characteristics of the system. The existing results are obtained as limiting cases from the present study.     

Investigating the thermal environment effects on geometrically nonlinear vibration of smart functionally graded plates

An analytical solution for a sandwich circular FGM plate coupled with piezoelectric layers under one-dimensional heat conduction is presented. All materials of the device may be of any functional gradients in the direction of thickness. The solution exactly satisfies all the equilibrium conditions and continuity conditions for the stress, displacement and electric displacement as well as electric potential on the interfaces between adjacency layers. A nonlinear static problem is solved first to determine the initial stress state and pre-vibration deformations of the FG plate that is subjected to in-plane forces and applied actuator voltage in thermal environment in the case of simply supported boundary conditions. By adding an incremental dynamic state to the pre-vibration state, the differential equations that govern the nonlinear vibration behavior of pre-stressed piezoelectric coupled FGM plates are derived. The role of thermal environment as well as control effects on nonlinear static deflections and natural frequencies imposed by the piezoelectric actuators using high input voltages are investigated. Numerical examples are provided and simulation results are discussed. Numerical results for FGM plates with a mixture of metal and ceramic are presented in dimensionless forms. The good agreement between the results of this paper and those of the finite element (FE) analyses validated the presented approach. In a parametric study the emphasis is placed on investigating the effect of varying the applied actuator voltage and thermal environment as well as gradient index of FG plate on the dynamics and control characteristics of the structure.     

An analytical solution for free vibration analysis of circular plates in axisymmetric modes based on the two variables refined plate theory

This paper presents, for the first time, an analytical solution for free vibrations of an isotropic circular plate in axisymmetric modes based on the two variables refined plate theory. This theory accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. Governing equations are derived using Hamilton’s principle and an analytical method on the basis of using Bessel functions is introduced to solve them. By this procedure, final form of the governing equations is obtained in matrix form. These equations are solved for classical boundary conditions and comparison studies are performed to verify the validity of the present results. It is found that the results obtained using RPT and TSDT are close to each other. As a benchmark, numerical results are presented in a dimensionless form for various values of thickness to radius ratio.     

小波伽辽金方法及其工程应用研究

研究了应用小波伽辽金方法求解一类具有奇异性的微分方程 ,给出了算法的通用格式。利用小波的两尺度关系和正交化条件 ,很好地解决了因 Daubechies小波尺度函数无明确解析表达式造成的积分困难 ,推导出了小波伽辽金变分格式及常用的系数矩阵。算法在边界条件的处理上 ,首先假设椭圆方程只满足周期边界条件 ,即源函数和方程右端项是周期函数 ,经过小波伽辽金变分后 ,得到了周期解的卷积形式 ,然后加入 Dirichlet边界条件 ,用容量矩阵法给出了局部区域上的解。如此处理 ,可以先分析方程的周期解 ,在对解有了粗略了解的基础上 ,选用较高阶的小波尺度函数对奇异点附近区域进行详细的计算 ,这样不但看出了微分方程的奇异性存在 ,而且给出方程的数值解。最后通过办公纸张瞬态温度场大梯度问题的数值分析 ,表明该方法具有满意的分析精度     

Simulation of 3-D Deformable Bodies Dynamics by Spectral Boundary Integral Equation Method

The study of the dynamics of inhomogeneous elastically deformable media under conditions of stationary and transient loading processes is based on the integral representation of the corresponding boundary value problems with time-dependent fundamental solutions and a combination of methods for solving boundary integral equations with Fourier transforms. Theoretical bases of methods for solving boundary integral equations focused on effective numerical implementation and application flexibility are presented. The accuracy of the proposed approach is discussed. Three-dimensional applied problems are solved in support of the effectiveness of the developed method.     

考虑温度效应的两端固支微机电开关梁静力分析

建立考虑温度效应的二维两端固支微机电开关梁静力变形分析模型,利用该模型分析温度变化引起的失稳问题,并利用该模型对两端固支微开关梁受电场力和温度作用下的受力变形进行分析。分析过程分为三个阶段：梁受电场力和温度变化作用,而在梁中点不受任何约束的变形阶段;除受电场力和温度变化作用外,梁中点变形受约束,因而梁中点受支反力作用的变形阶段;梁受电场力、温度变化以及支反力作用,并且在梁的中间一段具有指定挠度和转角的变形阶段。第二阶段中支反力的大小以及第三阶段中支反力的大小和位置都同驱动电压和温度变化有关。在实例中,给出部分计算结果,包括：温度变化对不同阶段梁受力变形的影响、对吸合电压的影响、对微开关电容的影响,以及不同温度变化下梁长对吸合电压的影响和不同温度变化下梁高对吸合电压的影响等,从中可以看出温度变化的作用及考虑温度效应的必要性。     

A boundary element solution approach for the conjugate heat transfer problem in thermally developing region of a thick walled pipe

This paper presents a sole application of boundary element method to the conjugate heat transfer problem of thermally developing laminar flow in a thick walled pipe when the fluid velocities are fully developed. Due to the coupled mechanism of heat conduction in the solid region and heat convection in the fluid region, two separate solutions in the solid and fluid regions are sought to match the solid-fluid interface continuity condition. In this method, the dual reciprocity boundary element method (DRBEM) with the axial direction marching scheme is used to solve the heat convection problem and the conventional boundary element method (BEM) of axisymmetric model is applied to solve the heat conduction problem. An iterative and numerically stable BEM solution algorithm is presented, which uses the coupled interface conditions explicitly instead of uncoupled conditions. Both the local convective heat transfer coefficient at solid-fluid interface and the local mean fluid temperature are initially guessed and updated as the unknown interface thermal conditions in the iterative solution procedure. Two examples imposing uniform temperature and heat flux boundary conditions are tested in thermally developing region and compared with analytic solutions where available. The benchmark test results are shown to be in good agreement with the analytic solutions for both examples with different boundary conditions.