Interfaces Between two Dissimilar Elastic Materials

In this paper the near tip solutions for interface corners written in terms of the stress intensity factors are presented in a unified expression. This single expression is applicable for any kinds of interface corners including corners and cracks in homogeneous materials as well as interface corners and interface cracks lying between two dissimilar materials, in which the materials can be any kinds of linear elastic anisotropic materials or piezoelectric materials. Through this unified expression of near tip solutions, the singular orders of stresses and their associated stress/electric intensity factors for different kinds of interface problems can be determined through the same formulae and solution techniques. This unified feature of solving interface problems is then implemented numerically through several different interface problems. Moreover, in order to improve the accuracy and efficiency of numerical computation, a special boundary element based upon the Green's function of bimaterials is introduced in this paper.     

含界面边裂纹压电材料反平面问题的应力强度因子

研究了含界面边裂纹的不同压电介质组成的复合材料在反平面荷载和平面内电场作用下的电弹场,得到了级数形式的基本解和应力强度因子,最后用边界配置法求解了应力强度因子。结果表明,在外加剪切荷载的作用下,应力强度因子与外加电场无关。     

A strong solution for a generalized plane strain unsymmetrical piezoelectric bi-layer laminates

Interfacial stresses, electric fields, and electric displacements of a piezoelectric unsymmetrical bi-layer orthotropic laminate are presented. A state space equation for orthotropic piezoelectric material is derived from three-dimensional piezoelectric elasticity directly. With the application of the transfer matrix and recursive solution approach, a strong solution for the unsymmetrical piezoelectric generalized plane strain bi-layer laminated structure is sought after considering all elastic and piezoelectric constants of materials. Electromechanical boundary layer effect is identified quantitatively at free edges. To facilitate the discussion on the results, the corresponding calculations from finite element models are compared with those of the strong solution.     

Three-dimensional BEM for piezoelectric fracture analysis

A boundary element approach with quadratic isoparametric elements, quarter-point elements and singular quarter-point elements for three-dimensional crack problems in piezoelectric solids under mechanical and electrical loading conditions, is presented in this paper for the first time. The procedure is based on Deeg's fundamental solution for anisotropic piezoelectric materials, and the classical extended displacement boundary integral equation. Stress and electric displacement intensity factors are directly evaluated as system unknowns, and also as functions of the computed nodal displacements and electric potentials at crack faces. Special attention is paid to the fundamental solution evaluation. Several three-dimensional crack problems in transversely isotropic bodies under mechanical and electrical loading conditions are analysed. Numerical solutions computed for prismatic cracked 3D plate problems with a plane strain behaviour are in very good agreement with their corresponding 2D BE solutions. Results for a penny shape crack in a piezoelectric cylinder are presented for the first time. The proposed approach is shown to be a simple, robust and useful tool for stress and electric displacement intensity factors evaluation in piezoelectric media.     

The boundary contour method for piezoelectric media with linear boundary elements

This paper presents a development of the boundary contour method (BCM) for piezoelectric media. First, the divergence‐free property of the integrand of the piezoelectric boundary element is proved. Secondly, the boundary contour method formulation is derived and potential functions are obtained by introducing linear shape functions and Green's functions (Computer Methods in Applied Mechanics and Engineering 1998; 158 : 65) for piezoelectric media. The BCM is applied to the problem of piezoelectric media. Finally, numerical solutions for illustrative examples are compared with exact ones and those of the conventional boundary element method (BEM). The numerical results of the BCM coincide very well with the exact solution, and the feasibility and efficiency of the method are verified. Copyright © 2003 John Wiley & Sons, Ltd.     

Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates

In this paper, nonlinear static and free vibration analysis of functionally graded piezoelectric plates has been carried out using finite element method under different sets of mechanical and electrical loadings. The plate with functionally graded piezoelectric material (FGPM) is assumed to be graded through the thickness by a simple power law distribution in terms of the volume fractions of the constituents. Only the geometrical nonlinearity has been taken into account and electric potential is assumed to be quadratic across the FGPM plate thickness. The governing equations are obtained using potential energy and Hamilton’s principle that includes elastic and piezoelectric effects. The finite element model is derived based on constitutive equation of piezoelectric material accounting for coupling between elasticity and electric effect using higher order plate elements. The present finite element is modeled with displacement components and electric potential as nodal degrees of freedom. Results are presented for two constituent FGPM plate under different mechanical boundary conditions. Numerical results for PZT-4/PZT-5H plate are given in dimensionless graphical forms. Effects of material composition and boundary conditions on nonlinear response are also studied. The numerical results obtained by the present model are in good agreement with the available solutions reported in the literature.     

The Boundary Contour Method for Piezoelectric Media with Quadratic Boundary Elements

This paper presents a development of the boundary contour method (BCM) for piezoelectric media. Firstly, the divergence-free of the integrand of the piezoelectric boundary element method is proved. Secondly, the boundary contour method formulations are obtained by introducing quadratic shape functions and Green's functions (Computer Methods in Applied Mechanics and Engineering1998;158: 65-80) for piezoelectric media and using the rigid body motion solution to regularize the BCM and avoid computation of the corner tensor. The BCM is applied to the problem of piezoelectric media. Finally, numerical solutions for illustrative examples are compared with exact ones. The numerical results of the BCM coincide very well with the exact solution, and the feasibility and efficiency of the method are verified.     

Numerical analysis for piezoelectric crack under varied boundary conditions by optimized hybrid element method

In this article, a piezoelectric hybrid element is presented and optimized by penalty equilibrium approach, and special crack surface element is suggested for exactly implementing the boundary conditions on crack surface. An iteration technique is used to treat one of the electric boundary conditions. Then, a piezoelectric material with crack is numerically studied by the optimized hybrid element method, and the results are compared with the analytical solutions. The stress and the electrical displacement fields with different crack surface conditions are studied, and the influence to those fields arisen by the far field mechanical and electric loading is also studied.     

A boundary element method for analysis of thermoelastic deformations in materials with temperature dependent properties

In this paper, the boundary element method (BEM) for solving quasi‐static uncoupled thermoelasticity problems in materials with temperature dependent properties is presented. The domain integral term, in the integral representation of the governing equation, is transformed to an equivalent boundary integral by means of the dual reciprocity method (DRM). The required particular solutions are derived and outlined. The method ensures numerically efficient analysis of thermoelastic deformations in an arbitrary geometry and loading conditions. The validity and the high accuracy of the formulation is demonstrated considering a series of examples. In all numerical tests, calculation results are compared with analytical and/or finite element method (FEM) solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

圆环形截面压电纤维复合材料有效电弹性性能研究

研究含周期分布压电纤维的压电复合材料的有效电弹性性能。通过在材料代表性体积单元边界上施加位移和电势周期边界条件,利用有限元法求得了代表性体积单元内的电弹性场。由平均电弹性场和压电复合材料有效电弹性性能定义,预测了圆环形截面压电纤维复合材料的有效电弹性系数。通过算例,比较了相同压电材料体积分数下圆环形截面压电纤维复合材料与圆截面压电纤维复合材料有效电弹性性能的差异,讨论了圆环形截面压电纤维内部非压电填充物的力学性质对有效压电系数的影响。该文结论可为高灵敏度压电复合材料设计提供 参考。     

Modeling of porous piezoelectric structures by the meshless local Petrov-Galerkin method

A meshless method based on the local Petrov-Galerkin approach is proposed to solve initial-boundary value problems of porous piezoelectric solids. Constitutive equations for porous piezoelectric materials possess a coupling between mechanical displacements and electric intensity vectors in both solid and fluid phases. Stationary and transient 2-D and 3-D axisymmetric problems are considered in this article. Nodal points are spread on the problem domain, and each node is surrounded by a small circle for simplicity. The spatial variation of displacements and electric potentials for both phases is approximated by the moving least-squares scheme. After performing the spatial integration, one obtains a system of ordinary differential equations for certain nodal unknowns. The resulting system is solved numerically by the Houbolt finite-difference scheme as a time stepping method. The proposed method is applied to bending problems associated with a porous piezoelectric 2-D plate and 3-D axisymmetric cylinder under simply supported and clamped boundary conditions.     

FEM analysis of electro-mechanical coupling effect of piezoelectric materials

Based on the mechanical and electrical equilibrium equations of piezoelectric materials, the minimum potential theory is presented by using the virtual work principle in this paper. A finite element method (FEM) formulation accounting for the electro-mechanical coupling effect of piezoelectric materials is given. Some problems in the numerical simulation are discussed and the extreme illness of the stiffness matrix is overcome by the dimension changing method. As a simple application, the response of an elliptical cavity in infinite media of piezoelectric materials is analyzed. Such a geometry leads to stress and electric field concentrations.     

A new finite‐element formulation for electromechanical boundary value problems

In this paper, a new finite‐element formulation for the solution of electromechanical boundary value problems is presented. As opposed to the standard formulation that uses scalar electric potential as nodal variables, this new formulation implements a vector potential from which components of electric displacement are derived. For linear piezoelectric materials with positive definite material moduli, the resulting finite‐element stiffness matrix from the vector potential formulation is also positive definite. If the material is non‐linear in a fashion characteristic of ferroelectric materials, it is demonstrated that a straightforward iterative solution procedure is unstable for the standard scalar potential formulation, but stable for the new vector potential formulation. Finally, the method is used to compute fields around a crack tip in an idealized non‐linear ferroelectric material, and results are compared to an analytical solution. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical computation of the electromagnetic field in the electrodes of a three‐phase arc furnace

In this paper we introduce and numerically solve a mathematical model for numerical simulation of electro‐magnetic field in a three‐phase electric reduction furnace. The model allows us to compute the current distribution on a cross‐section of the three electrodes. A combined boundary element/finite element method is used. Numerical results for real industrial furnaces are shown. As a by‐product we compute the torque on the electrodes due to the Lorentz electromagnetic force. Copyright © 1999 John Wiley & Sons, Ltd.     

Combined analytical and numerical solution of 2D interface corner configurations between dissimilar piezoelectric materials

The failure assessment of smart composite structures requires efficient analytical and numerical techniques in order to tackle electrical and mechanical field concentrations. The present work is directed to the analysis of interface corner and crack configurations which occur in smart composite materials. It delivers a new technique to solve the corresponding piezoelectric boundary value problems. The purpose of the given paper is to describe exactly the asymptotic behaviour at piezoelectric interface corner configurations using the eigenfunction expansions on the one hand, and in the linking of these expansions to regular finite elements on the other. Specific singular eigenfunctions for homogeneous and interface crack configurations are discussed. For the considered cases, the classical crack modes (Mode I and Mode II) and a new Electric Mode are identified. The coupling of the full eigenfunction expansions to the finite elements surrounding the tip region is based on the principle of virtual work applied to the orthogonalised eigenfunctions. Finally, one gets an asymptotic stiffness matrix which does not depend on the distance to the tip. The coefficients of the eigenfunctions can be obtained efficiently from the generalised displacements of the global solution by means of the orthogonalised eigenfunctions. The technique allows to numerically bypass possible singular oscillatory terms in the weak sense, although they actually exist in the strong solution. The given approach is proven and verified in numerical test examples. Standard finite element methods encounter difficulties to give correct solutions at piezoelectric interface crack tips.     

Boundary element‐free method for fracture analysis of 2‐D anisotropic piezoelectric solids

This paper considers a 2‐D fracture analysis of anisotropic piezoelectric solids by a boundary element‐free method. A traction boundary integral equation (BIE) that only involves the singular terms of order 1/r is first derived using integration by parts. New variables, namely, the tangential derivative of the extended displacement (the extended displacement density) for the general boundary and the tangential derivative of the extended crack opening displacement (the extended displacement dislocation density), are introduced to the equation so that solution to curved crack problems is possible. This resulted equation can be directly applied to general boundary and crack surface, and no separate treatments are necessary for the upper and lower surfaces of the crack. The extended displacement dislocation densities on the crack surface are expressed as the product of the characteristic terms and unknown weight functions, and the unknown weight functions are modelled using the moving least‐squares (MLS) approximation. The numerical scheme of the boundary element‐free method is established, and an effective numerical procedure is adopted to evaluate the singular integrals. The extended ‘stress intensity factors’ (SIFs) are computed for some selected example problems that contain straight or curved cracks, and good numerical results are obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

Coupled three‐layered analysis of smart piezoelectric beams with different electric boundary conditions

This paper presents a theoretical and finite element (FE) formulation of a three‐layered smart beam with two piezoelectric layers acting as sensors or actuators. For the definition of the mechanical model a partial layerwise theory is considered for the approximation of the displacement field of the core and piezoelectric face layers. An electrical model for different electric boundary conditions (EBC), namely, electroded layers with either closed‐ or open‐circuit electrodes with electric potential prescribed or layers without electrodes, is considered. Using a variational formulation, the direct piezoelectric effect for the different EBC is physically incorporated into the mechanical model through appropriate approximations of the electric field in the axial and transverse directions. An FE model of a three‐layered smart beam with different EBC is proposed considering a fully coupled electro‐mechanical theory through the use of effective stiffness parameters and a modified static condensation. FE solutions of the quasi‐static electrical and mechanical actuations and natural frequencies are presented. Comparisons with numerical FE and analytical solutions available in the literature demonstrate the representativeness of the developed theory and the effectiveness of the proposed FE model for different EBC. Copyright © 2005 John Wiley & Sons, Ltd.     

Dynamic response for non-homogeneous piezoelectric medium with multiple cracks

A general procedure to analyze the dynamic response of non-homogeneous piezoelectric medium containing some non-collinear cracks is developed. It is assumed that all the material properties only depend on the coordinates y (along the thickness direction). The assumption is made that the non-homogeneous medium is composed of numerous laminae with their surfaces perpendicular to the thick direction. The solution method is based upon the Fourier and Laplace transforms to reduce the boundary value problem to a system of generalized singularity equations in the Laplace transform domain. The singular integral equations for the problem are derived and numerically solved by weight residual value method. The time-dependent full field solutions are obtained in the time domain. As numerical illustration, the stress and electric displacement intensity factors for a three-layer plate specimen with two cracks are presented. It is found that the stress and electric fields are coupled in the crack plane ahead of the crack tip for non-homogenous piezoelectric materials.     

Exact solution for functionally graded and layered magneto-electro-elastic plates

In this paper, an exact solution is presented for the multilayered rectangular plate made of functionally graded, anisotropic, and linear magneto-electro-elastic materials. While the edges of the plate are under simply supported conditions, general mechanical, electric and magnetic boundary conditions can be applied on both the top and bottom surfaces of the plate. The functionally graded material is assumed to be exponential in the thickness direction and the homogeneous solution in each layer is obtained based on the pseudo-Stroh formalism. For multilayered plate structure, the propagator matrix method is employed so that only a 5 × 5 system of linear algebraic equations needs to be solved. The exact solution is then applied to two functionally graded (exponential) sandwich plates made of piezoelectric BaTiO₃ and magnetostrictive CoFe₂O₄, under mechanical and electric loads on the top surface. While the numerical results clearly show the influence of the exponential factor, magneto-electro-elastic properties, and loading types on induced magneto-electric-elastic fields, they can also serve as benchmarks to numerical methods such as the finite and boundary element methods.     

Analysis of cracked piezoelectric solids by a mixed three-dimensional BE approach

This paper has two main objectives in relation to the analysis of three-dimensional crack problems in piezoelectric solids. The first one is to present the formulation, effective implementation and numerical treatment of a mixed boundary element technique for the study of this type of problems. The numerical procedure is based on the use of extended displacement and extended traction integral equations for external and crack boundaries, respectively. The boundary element formulation is presented with particular emphasis on numerical aspects related to singular kernels regularization and evaluation of boundary integrals. Quadratic boundary elements and quarter-point boundary elements are implemented in a computer code. By using these elements, electric and stress intensity factors are directly computed from nodal values at quarter-point elements. The second purpose is to study several realistic piezoelectric crack problems for the first time. Unbounded and bounded cracked piezoelectric three-dimensional (3D) solids with different geometries are studied. Results presented in this paper can be used as a reference for future research. Prior to the analysis of problems whose solution was previously unknown, the technique is validated by solving some simple problems with known analytical or numerical solution. Then, more realistic crack problems of engineering interest have been analysed for the first time. In all cases, results for the solid deformed shape, the crack opening displacements and the extended stress intensity factor components, are shown.