On the singularities of the thermo-electro-elastic fields near the apex of a piezoelectric bonded wedge

Based on the generalized Lekhnitskii formulation and Mellin transform, the thermo-electro-elastic fields of a piezoelectric bonded wedge are investigated in this paper. From the potential theory in a wedge-shaped region, a general form of the temperature change is proposed as a particular solution in the generalized Lekhnitskii formulation. The emphasis is on the singular behavior near the apex of the piezoelectric bonded wedge, including singularity orders and angular functions, which can be computed numerically. The interface between two materials can be either perfectly bonded, namely type A, so that the continuity of electric displacements holds, or a thin electrode, namely type B, so that the electric potential is grounded. Case studies of PZT-5H/PZT-4 and graphite-epoxy/PZT-4 bonded wedges reveal that, in most cases, the type B continuity condition has more severe singularities than type A due to the mixed boundary point of the electrostatics at the apex of the wedge. The results of this study show that the reduction or disappearance of singularity orders is possible through the appropriate selection of poling/fiber orientations and wedge angles.     

Stress singularities near the apex of a cylindrically polarized piezoelectric wedge

Summary In this paper, the stress singularities for a cylindrically polarized piezoelectric wedge are investigated. The characteristic equations are derived analytically by using the extended Lekhnitskii formulation. The piezoelectric material (PZT-4) is polarized in the radial, circular or axial direction, respectively. Similar to the rectilinearly polarized piezoelectric problem, the inplane and antiplane stress fields are uncoupled. The results show the variations of the singularity orders with the changes of the wedge angle, material constants, polarized direction, and the boundary conditions.     

Antiplane electro-mechanical field of a piezoelectric finite wedge under shear loading and at fixed-grounded boundary conditions

This paper presents the general solutions of antiplane electro-mechanical field solutions for a piezoelectric finite wedge subjected to a pair of concentrated forces and free charges. The boundary conditions on the circular segment are considered as fixed and grounded. Employing the finite Mellin transform method, the stress and electrical displacement at all fields of the piezoelectric finite wedge are derived analytically. In addition, the singularity orders and intensity factors of stress and electrical displacement can also be obtained. These parameters can be applied to examine the fracture behavior of the wedge structure. After being reduced to the problem of an antiplane edge crack or an infinite wedge in a piezoelectric medium, the results compare well with those of previous studies.     

Electroelastic behavior of doubly periodic piezoelectric fiber composites under antiplane shear

The piezoelectric composites with a doubly periodic parallelogrammic array of piezoelectric fibers are dealt with under antiplane shear coupled with inplane electrical load. A rigorous analytical method is developed by using the doubly quasi-periodic Riemann boundary value problem theory integrated with the eigenstrain and eigen-electrical-field concepts. The numerical results are presented and a comparison with finite element calculations, experimental data and micromechanical analysis is made to demonstrate the efficiency and accuracy of the present method. Subsequently, the present solutions are used to study two important topics in piezoelectric fiber composites, i.e., (1) stress and electrical field fluctuations in the microstructure, (2) the macroscopic effective electroelastic moduli. The relation between the macroscopic properties of the composites and their microstructural parameters is discussed and many interesting electroelastic interaction phenomena are revealed, which are useful to estimate and optimize the performance of piezoelectric composites.     

Antiplane stress singularities in a bonded bimaterial piezoelectric wedge

Summary  In this paper, the eigen-equations governing antiplane stress singularities in a bonded piezoelectric wedge are derived analytically. Boundary conditions are set as various combinations of traction-free, clamped, electrically open and electrically closed ones. Application of the Mellin transform to the stress/electric displacement function or displacement/electric potential function and particular boundary and continuity conditions yields identical eigen-equations. All of the analytical results are tabulated. It is found that the singularity orders of a bonded bimaterial piezoelectric wedge may be complex, as opposed to those of the antiplane elastic bonded wedge, which are always real. For a single piezoelectric wedge, the eigen-equations are independent of material constants, and the eigenvalues are all real, except in the case of the combination C–D. In this special case, C–D, the real part of the complex eigenvalues is not dependent on material constants, while the imaginary part is. Received 26 March 2002; accepted for publication 2 July 2002     

Penny-shaped crack in a piezoelectric cylinder under electro-mechanical loads

Summary The fracture behavior of a penny-shaped crack in a axisymmetrical piezoelectric ceramic cylinder of finite radius under mechanical and electrical loads is analyzed under electric continuous boundary conditions on the crack surface. The potential theory and Hankel transform are used to obtain a system of dual integral equations, which is then expressed as a Fredholm integral equation. Singular mechanical and electrical fields and field intensity factors of mode I are obtained. The numerical values of various field intensity factors for PZT-6B piezoelectric ceramics are graphically shown for a uniform load and a ring-shaped load, respectively. The effects of the radius of the cylinder on the field intensity factors are investigated. This work was supported by the Korea Research Foundation Grant (KRF-2001-041-E00057).     

Elastic-viscoplastic fields near the tip of a propagating crack under anti-plane shear

The elastic-viscoplastic model proposed by Bingham was used to analyse the stress and strain surrounding the tip of a propagating crack under antiplane shear. The proper displacement pattern was given ; the asymptotic equations were derived and solved numerically. The analysis and calculation show that for smaller viscosity the crack-tip possesses logarthmic singularity, and for larger viscosity it possesses power-law singularity.In critical case, the two kinds of singularity are consistent with each other. The result revealed the important role of viscosity for crack-tip field.     

Angle cracks in two bonded functionally graded piezoelectric material under anti-plane shear

Consider two bonded functionally graded piezoelectric material (FGPM) with finite height. Each material contains an arbitrary oriented crack. The material properties are assumed in exponential forms in the direction normal to the interface. The crack surface condition is assumed to be electrically impermeable or permeable. Using the Fourier transform technique, the problem can be reduced to a system of singular integral equations, which are then solved numerically by applying the Gauss-Chebyshev integration formula to obtain the stress intensity factors at the crack tips. Numerical calculations are carried out to obtain the energy density factor S and the energy release rate G. In impermeable case, the energy release rate has been shown to be negative as the electric loads are applied. The positive definite characteristic of the energy density factor makes it possible for predicting the fracture behavior of the cracked structure. The influences of the non-homogeneous parameters and crack orientation on the energy density factors at the crack tips are discussed in detail. The results show that the energy density factor at the crack tip will be increased when the crack tip is located within the softer material.     

Dynamic response of a cracked piezoelectric ceramic under arbitrary electro-mechanical impact

The dynamic response of a cracked piezoelectric ceramic under in-plane electric and anti-plane mechanical impact is investigated by the integral transform method. The electric and mechanical loads are assumed to be arbitrary functions of time. It is shown that the dynamic crack-tip stress and electric displacement fields still have a square-root singularity. Numerical computations for the dynamic stress intensity factor show that the electric load has a significant influence on the dynamic response of stress field. On the other hand, the dynamic response of the electric field is determined solely by the applied electric field. The electric field will promote or retard the propagation of the crack depending on the time elapsed since the application of the external electro-mechanical loads.     

Transient response of a cracked piezoelectric strip under arbitrary electro-mechanical impact

By using the well-developed integral transform methodology, the dynamic response of stress and electric displacement around a finite crack in an infinite piezoelectric strip are investigated under arbitrary dynamic anti-plane loads. The dynamic stress intensity factors and electric displacement are obtained analytically. It is shown that the dynamic crack-tip stress and electric field still have a square-root singularity. Numerical computations for the dynamic stress intensity factor show that the electric load has a significant influence on the dynamic response of stress field. The higher the ratio of the crack length to the width of the strip, the higher the peak value of the dynamic stress intensity factor is. On the other hand, the dynamic response of the electric field is determined solely by the applied electric load. The electric field will promote or retard the propagation of the crack depending on the time elapse since the application of the external electro-mechanical loads. The project supported by the National Natural Science Foundation of China and the Post-Doctor Science Foundation of China     

Multi-interface cracks in a piezoelectric layer bonded to two half-piezoelectric spaces under anti-plane shear loading

The behavior of four parallel symmetry permeable interface cracks in a piezoelectric layer bonded to two half-piezoelectric spaces under anti-plane shear loading is investigated. By using the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations. These equations are solved by the Schmidt method. This process is quite different from that papers adopted previously. The normalized stress and electrical displacement intensity factors are determined for different geometric and property parameters for permeable crack surface conditions. Numerical examples are provided to show the effect of the geometry of the interacting cracks, the thickness and the materials constants of the piezoelectric layer upon the stress and electric displacement intensity factors of the cracks. It is found that the electric displacement intensity factors for the permeable crack surface conditions are much smaller than the results for the impermeable crack surface conditions.     

Elastodynamic analysis of a crack at an arbitrary angle to the graded interfacial zone in bonded half-planes under antiplane shear impact

A solution is provided for the elastodynamic problem of a crack at an arbitrary angle to the graded interfacial zone in bonded media under the action of antiplane shear impact. The interfacial zone is modeled by a nonhomogeneous interlayer with the spatially varying shear modulus and mass density in terms of power functions between the two dissimilar, homogeneous half-planes. Based on the use of Laplace and Fourier integral transforms and the coordinate transformations of basic field variables, formulation of the transient crack problem is reduced to solving a Cauchy-type singular integral equation in the Laplace transform domain. The crack-tip response in the physical domain is recovered via the inverse Laplace transform and the values of dynamic mode III stress intensity factors are obtained as a function of time. A comprehensive parametric study is then presented of the effects of crack obliquity on the overshoot behavior of the transient crack-tip response, by plotting the peak values of the dynamic stress intensity factors versus the crack orientation angle for various material and geometric combinations of the bonded system.     

Singularities of physical fields at the apex of a piezoceramic wedge (plane strain)

A uniform piezoceramic wedge is studied. Singularities of coupled physical fields at the apex of the wedge are investigated with homogeneous boundary conditions on its faces. The problem is solved by using complex representations of the components of the electroelastic field for two basic types of mechanical and electrical boundary conditions. The results of computations are presented. Sumy State University, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 3, pp. 89–92, March, 1999.     

Singular plastic fields in wedge indentation of pressure sensitive solids

The paper examines singular plastic fields induced near the tip of a wedge indentating a pressure sensitive solid. Plane strain conditions are assumed and material response is modelled by the small strain Drucker–Prager rigid/plastic constitutive law. A standard separation of variables solution is numerically generated for pure power-law hardening. Three possible measures of wall roughness are studied with an attempt to expose the coupling between wall friction and material pressure sensitivity. Sample calculations illustrate that stress singularity decreases with increasing friction, wedge angle and hardening exponent, but increases with pressure sensitivity. At large values of the hardening exponent, when the material is nearly perfectly plastic, effective stress contours approach the slip line limit. The concept of indentation index is introduced as a possible estimate for average indentation pressure.     

The analysis of thermal residual stresses near the apex in bonded dissimilar materials

By employing the complex variable method and constructing the particular solution sequences in the form of complex functions, all the cases of the thermal residual stress field near the apex in dissimilar materials bonded with two arbitrary angles are researched theoretically, and the corresponding classical solutions are obtained. Moreover, the primary paradox, the secondary paradox and even the triple paradox are discovered in the classical solutions and also resolved here, thereby it is confirmed that thermal residual stresses near the apex in bonded dissimilar materials probably possess the singularities of lnr (when the primary paradox occurs) , ln²r (when the secondary paradox occurs) and even ln³r (when the triple paradox occurs) . In addition, the systematic method to solve multiple paradox problems is put forward. © 1999 Elsevier Science Ltd. All rights reserved.     

A unified treatment of the elastic elliptical inclusion under antiplane shear

Summary A generalized and unified treatment is presented for the antiplane problem of an elastic elliptical inclusion undergoing uniform eigenstrains and subjected to arbitrary loading in the surrounding matrix. The general solution to the problem is obtained through the use of conformal mapping technique and Laurent series expansion of the associated complex potentials. The resulting elastic fields are derived explicitly in both transformed and physical planes for the inclusion and the surrounding matrix. These relations are universal in the sense of being independent of any particular loading as well as the geometry of the matrix. The complete field solutions are provided for an elliptical inclusion under uniform loading at inifinity, and for a screw dislocation interacting with the elastic elliptical inclusion.     

Crack spacing effect for a piezoelectric cylinder under electro-mechanical loading or transient heating

This paper investigates the fracture problem of a piezoelectric cylinder with a periodic array of embedded circular cracks. An electro-mechanical fracture mechanics model is established first. The model is further used to the thermal fracture analysis of a piezoelectric cylinder subjected to a sudden heating on its outer surface. The temperature field and the associated thermal stresses and electric displacements are obtained and are added to the crack surface to form a mixed-mode boundary value problem for the electro-mechanical coupling fracture. The stress and stress intensities are investigated for the effect of crack spacing. Strength evaluation of piezoelectric materials under the transient thermal environment is made and thermal shock resistance of the medium is given.     

Contact problem for a plane crack under a normally incident antiplane shear wave

The contact-interaction problem for a stationary plane crack with friction between its edges under the action of a normal (to the crack plane) harmonic shear wave is addressed. Antiplane deformation conditions are considered. The distribution of contact forces and displacement discontinuity of crack edges are studied Published in Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 138–142, May 2007.