Piezoelectric hollow cylinder with thermal gradient

The coupling nature of piezoelectric materials has acquired wide applications in electric-mechanical and electric devices. 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, an analytic solution to the axisymmetric problem of a radially polarized, radially orthotropic piezoelectric hollow cylinder with thermal gradient is developed. 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. The stress and potential field distributions in the cylinder are obtained numerically for two piezoceramics. It is shown that the hoop stresses in a cylinder composed of these materials can be decreased throughout the cross-section by applying an appropriate set of boundary conditions. This paper was recommended for publication in revised form by Associate Editor Jeong Sam Han Mahdi Saadatfar received a B. S. degree in Mechanical Engineering from University of Kashan 2006. He is currently a M.S student at the School of Mechanical Engineering at University of Tehran, Iran. He is currently researching about modeling of nanoindentation process in nanocomposites. Mr. Saadatfar’s research interests are in the area of piezoelectric Materials, Polymer/Clay nanocomposites and Finite element modeling. He has several published paper about piezoelectric materials and Finite element modeling of nanocomposites. Amin Shariat Razavi received a B.S degree in Mechanical Engineering from Kashan University in 2006. He is currently testing and examining an specific type of intelligent plasma cutting machine for process equipment that is designed by himself. Mr. Razavi’s research interests are smart materials and design of mechanical system.     

Nonlinear viscoelastic analysis of thick-walled cylindrical composite pipes

This paper analyzes the effect of the polymer matrix non-viscoelastic behaviour in the mechanical behaviour of thick multilayered cylinders. The original contribution of this work is to provide novel approximate analytical solutions to compute the time-dependent internal stress state through the pipe thickness within the framework of nonlinear viscoelasticity theory. The structures considered are thick, multilayered anisotropic infinitive long cylinders subjected to axisymmetric mechanical loading. Under such conditions there is an exact elastic solution which naturally satisfies equilibrium, strain-displacement, compatibility and boundary conditions for the stated constitutive equations and loading. Due to the continuous stress variations through the cylinder thickness, the proposed nonlinear viscoelastic solution assumes the averaged stress state to calculate the nonlinear elastic and viscoelastic factors in each layer. Furthermore, the solution is obtained assuming that the creep strains, within each layer, are constant through the thickness. The proposed algorithm converges to the exact solution when the number of layers is artificially increased. For the linear viscoelastic case, the proposed solution proved to match the exact known solution for isotropic viscoelastic materials. Finally several invented cases are run to illustrate the importance of the viscoelasticity phenomenon on the internal stress field in thick-laminated cylinders.     

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.     

Dynamic responses of a multilayered pyroelectric hollow cylinder of crystal class 2 mm for axisymmetric plane strain problems

The dynamic solution of a multilayered pyroelectric hollow cylinder of crystal class 2 mm in the state of axisymmetric plane strain is obtained. By the principle of superposition, the solution is divided into two parts: One is quasi-static and the other is dynamic. The quasi-static part is obtained in an explicit form by the state space method, and the dynamic part is derived by the separation of variables method coupled with the initial parameter method as well as the orthogonal expansion technique. By using the obtained quasi-static and dynamic parts and the electric boundary conditions as well as the electric continuity conditions, a Volterra integral equation of the second kind with respect to a function of time is derived, which can be solved successfully by means of the interpolation method. The displacements, stresses and electric potentials can be finally determined. The present method is suitable for a multilayered pyroelectric hollow cylinder of crystal class 2 mm consisting of arbitrary layers and subjected to arbitrary axisymmetric thermal loads. Numerical results are finally presented and discussed.     

Shape design sensitivity analysis for interface problem in axisymmetric elasticity

A boundary integral equation method in the shape design sensitivity analysis is developed for the elasticity problems with axisymmetric non-homogeneous bodies. Functionals involving displacements and tractions at the zonal interface are considered. Sensitivity formula in terms of the interface shape variation is then derived by taking derivative of the boundary integral identity. Adjoint problem is defined such that displacement and traction discontinuity is imposed at the interface. Analytic example for a compound cylinder is taken to show the validity of the derived sensitivity formula. In the numerical implementation, solutions at the interface for the primal and adjoint system are used for the sensitivity. While the BEM is a natural tool for the solution, more generalization should be made since it should handle the jump conditions at the interface. Accuracy of the sensitivity is evaluated numerically by the same compound cylinder problem. The endosseous implant-bone interface problem is considered next as a practical application, in which the stress value is of great importance for successful osseointegration at the interface. As a preliminary step, a simple model with tapered cylinder is considered in this paper. Numerical accuracy is shown to be excellent which promises that the method can be used as an efficient and reliable tool in the optimization procedure for the implant design. Though only the axisymmetric problem is considered here, the method can be applied to general elasticity problems having interface.     

The unified equations to obtain the exact solutions of piezoelectric plane beam subjected to arbitrary loads

The unified equations to obtain the exact solutions for piezoelectric plane beam subjected to arbitrary mechanical and electrical loads with various ends supported conditions is founded by solving functional equations. Comparing this general method with traditional trial-and-error method, the most advantage is it can obtain the exact solutions directly and does not need to guess and modify the form of stress function or electric displacement function repeatedly. Firstly, the governing equation for piezoelectric plane beam is derived. The general solution for the governing equation is expressed by six unknown functions. Secondly, in terms of boundary conditions of the two longitudinal sides of the beam, six functional equations are yielded. These equations are simplified to derive the unified equations to solve the boundary value problems of piezoelectric plane beam. Finally, several examples show the correctness and generalization of this method.     

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.     

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.     

Analytical stress intensity factors for edge-cracked cylinder

We use the Hamiltonian formalism in elasticity to analyze edge-cracked cylinder under various three-dimensional loading conditions. The Hamiltonian form enables the successful separation of the independent variables in polar coordinates so that symplectic eigenfunctions can be analytically determined. The displacements and stresses are proved to be conjugating to each other and can be expanded in series of the symplectic eigenfunctions. The coefficients of the series are determined from the lateral boundary conditions along the crack faces and the outer boundary conditions along the finite geometric domain. The stress intensity factors and T-stresses near the crack-tip are found analytically. Three modes of stress intensity factors can be obtained simultaneously. The result indicates that the stress intensity factors depend directly on the respective first few coefficients of the general eigenvalue solutions. Examples for mixed boundary conditions, e.g., partly clamped and partly forced, are included. The influence of various parameters on the stress intensity factors is discussed. Since the method is analytic, the results can be considered as benchmark for numerical methods in determining singularities.     

State space solution for thick laminated piezoelectric plates with clamped and electric open-circuited boundary conditions

Based on the theories of 3-D elasticity and piezoelectricity and by assuming appropriate boundary functions, the state equation for laminated piezoelectric plate is established. By using the transfer matrix and recursive solution approach, an analytical solution that satisfies all boundary conditions, including the conditions on the top and bottom surfaces, of the laminates is presented. The solution can take into account all the independent elastic and piezoelectric constants for orthotropic and piezoelectric materials and satisfies the continuity conditions between plies of the laminates. Numerical examples are given at the end of the paper to verify the effectiveness of the present method. The results are compared with those of existing analytical and finite element models.     

Some mixed boundary value problems for the semi-infinite elastic solid subjected to tangential surface tractions

A method is developed for representing the tangential displacements and tractions at the surface of the semi-infinite solid in terms of potential functions. In this form, a mathematical analogy is revealed between corresponding mixed boundary value problems involving tangential and normal surface displacements respectively. This analogy enables a general solution to be obtained to the problem in which the surface tangential displacements are specified axisymmetric functions inside the circle ar0 and the tangential surface traction is zero outside this circle. The method can also be used for certain non-axisymmetric problems, but it fails if the indentation analogue has a stress singularity at the boundary of the stressed area.     

捞油抽子密封橡胶筒力学性能研究

密封橡胶筒的形变、应力以及其受油管内壁摩擦力对采油漏失率和使用寿命有着直接影响.采用弹性力学中的拉梅解答经过坐标变换得到轴向应力函数表达式,再通过密封橡胶筒边界条件和物理定律建立与油管内壁受力关系.得到橡胶筒形变、应力状态、摩擦力之间的函数关系,这种函数关系将有助于对井下捞油抽子进行定量研究.     

Advanced inelastic analysis of solids by the boundary element method

Two boundary element solution algorithms are used for solving a range of elastoplastic and thermoplastic problems. The first algorithm is the conventional iterative procedure in which the unknown boundary solution and the initial stress (or strain) rates are found together in an incremental iterative fashion. The second algorithm is a new variable stiffness type approach in which the incremental boundary solution is obtained in a direct (non-iterative) manner. This new approach is presented in a general manner for axisymmetric, two- and three-dimensional analyses. The formulation is implemented in a general purpose, multi-region system that utilizes quadratic isoparametric shape functions to model the geometry and field variables of the body and can admit up to 15 substructured regions of different material properties.     

Thermoelastic stability of duplex heat exchanger tubes

When duplex tubes are used for heat exchange purposes, thermal distortion affects the contact pressure and, hence, the thermal contact resistance at the interface. The resulting coupling between the thermal and thermoelastic fields for the system can lead to instability and non-uniqueness of the steady-state solution and, hence, to erratic heat transfer performance.Stability of the system is investigated by determining the conditions under which a small (axisymmetric or non-axisymmetric) perturbation on the steady state can grow exponentially in time. Substitution into the governing heat conduction and thermoelastic equations enables the unknown functions to be determined except for a set of arbitrary constants, which are then determined from the thermal and mechanical boundary conditions, including a statement of the relation between thermal contact resistance and pressure, linearized for small perturbations about the steady state.Results are presented for a range of material combinations and for both directions of steady state heat flux. It is shown that unstable axisymmetric perturbations can only occur for inward heat flow and, hence, the stability boundary for outward heat flow is always associated with a non-axisymmetric mode. In the latter case, the circumferential wavelength of the critical mode is generally about twice the tube thickness and a good approximation to the stability boundary can be obtained using previously published results for the stability of two plane layers in thermoelastic contact.For inward heat flow, the critical heat flux for axisymmetric perturbations is independent of the mean contact resistance, whereas that for non-axisymmetric modes increases with contact resistance. Thus, the stability boundary in this case is determined by non-axisymmetric modes for small values of contact resistance and by axisymmetric modes for large contact resistance.     

Refined analysis of axi-symmetric circular cylinder in the axial magnetic field without ad hoc assumption

The refined theory for axi-symmetric magnetoelastic circular cylinder is deduced systematically and directly from linear magnetoelasticity theory. Based on the general solution of magnetoelastic equation and the Lur’e method, the refined theory yields the solutions for magnetoelastic circular cylinder without ad hoc assumptions. On the basis of the refined theory developed in the present study, solutions are obtained for magnetoelastic circular cylinder with homogeneous and non-homogenous boundary conditions, respectively. For the circular cylinder with homogeneous boundary conditions, the refined theory provides exact solutions that satisfy all of the governing equations. The exact solutions can be divided into three parts: the 2-orders equation, the transcendental equation, and the magnetic equation. In the case of non-homogenous boundary conditions, the approximate governing equations are accurate up to the high-order terms with respect to cylinder radius.     

Couette flows of rigid/plastic solids: analytical examples of the interaction of constitutive and frictional laws

The exact analytical solution under plane strain conditions is found for a hollow circular cylinder subjected to rotational friction on the inner surface and an uniform pressure on the outer surface. Two constitutive laws, a rigid/plastic hardening model with a saturation stress (Voce-Palm material) and a rigid/viscoplastic model (Bingham material), are considered as well as three interfacial laws, sticking, Coulomb sliding friction and Tresca shearing friction. The study focuses on general behavior of the solutions for various pairs of constitutive and interfacial models, such as existence and uniqueness and qualitative differences in solution. For the Voce-Palm material, it is shown that solutions do not exist for certain boundary conditions if only sticking or Coulomb friction are permitted. On the other hand, multiple solutions exist for certain boundary conditions if only sticking and Tresca friction are permitted. Existence is achieved under all boundary conditions if sticking and both frictional laws permitted simultaneously. Uniqueness and unambiguity are achieved if two interface selection principles are invoked: (1) sticking must occur if it is possible and (2) Coulomb sliding must occur if it is possible and sticking is not. Geometrical interpretation of results in the form of friction maps is provided to illustrate possible regimes of interfacial behavior for different boundary conditions.     

An exact analytical solution for the self-weight deflection of circular plates of constant thickness

An exact analytical solution, not previously noted in the literature, for the self-weight deflection of a circular plate is obtained by superposition of various elementary solutions due to Love, and the accuracy of the existing approximate procedures is examined. The expressions for stress components are obtained using the axisymmetric stress-strain relations and the validity of the derived solution is justified by satisfying the field equations and the required boundary conditions.     

带硬涂层圆盘构件的固有振动特性的精确解法

研究了在广义弹性简支边界条件下的具有硬涂层的圆盘构件的自由振动的量纲一固有频率的精确解.首先利用多铁性多层圆盘的解析分析的多层板弹性理论,导出带硬涂层的圆盘结构的状态方程,其中以位移、电势、磁势、应力、电位移和磁感应强度为状态变量.利用有限Hankel变换和传播矩阵法,得到考虑压电和压磁效应的带硬涂层的圆盘的量纲一固有频率的精确解.根据算例结果,比较了压电、压磁两类硬涂层材料在单面涂层、双面涂层和不同涂层厚度的结构配置下的固有频率变化规律.     

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.     

Prediction of dimensional instability resulting from layer removal of an internally stressed orthotropic composite cylinder

When a layer of cylindrical composite component containing an axisymmetric residual stress state is removed from the inner or outer surface, the dimension of the remaining material changes to balance internal forces. Therefore, in order to machine cylindrical composite components within tolerances, it is important to know dimensional changes caused by residual stress redistribution in the body. In this study, analytical solutions for dimensional changes and the redistribution of residual stresses due to the layer removal from a residually stressed cylindrically orthotropic cylinder were developed. The cylinder was assumed to have axisymmetric radial, tangential and axial residual stresses. The result of this study is useful in cases where the initial residual stress distribution in the component has been measured by a non-destructive technique such as neutron diffraction with no information on the effect of layer removal operation on the dimensional changes.