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).     

Crack energy density of a piezoelectric material under general electromechanical loading

Crack energy density is considered and used as a possible fracture parameter in piezoelectricity under arbitrary electromechanical remote loads. The closed-form solution of a crack in a piezoelectric infinite plate subjected to general static electromechanical loading is obtained through a method alternative to the more common Stroh’s formalism. This analytical method, which is based on the spectral theorem of linear algebra, involves a transformation of similarity induced by the fundamental matrix in order to express the equations governing the problem in terms of complex potentials. The application of the mechanical boundary condition of stress-free crack and of one of the three considered electric boundary conditions (impermeable, permeable or semipermeable) leads then to the formulation of a Hilbert problem whose solution yields the stress and displacement fields. The crack energy density factors for mixed mode are then calculated under different mechanical and electrical loadings, as well as under different electric boundary conditions. The non-singular terms of the stress expressions are retained as well. The definition of the minimum energy density fracture criterion, as proposed by Sih, is given, and the influence of load biaxiality and positive or negative applied electric field on the criterion results is analyzed. The prediction of the incipient branching angle as from the energy density approach is also compared to that arising from the maximum circumferential stress theory for a mixed mode loading condition. Numerical results and graphs are presented and discussed for a PZT-4 piezoelectric ceramic.     

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.     

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     

Single-frequency vibrations and vibrational heating of a piezoelectric circular sandwich plate under monoharmonic electromechanical loading

The forced monoharmonic bending vibrations and dissipative heating of a piezoelectric circular sandwich plate under monoharmonic mechanical and electrical loading are studied. The core layer is passive and viscoelastic. The face layers (actuators) are piezoelectric and oppositely polarized over the thickness. The plate is subjected to harmonic pressure and electrical potential. The viscoelastic behavior of the materials is described by complex moduli dependent on the temperature of heating. The coupled nonlinear problem is solved numerically. A numerical analysis demonstrates that the natural frequency, amplitude of vibrations, mechanical stresses, and temperature of dissipative heating can be controlled by changing the area and thickness of the actuator. It is shown that the temperature dependence of the complex moduli do not affect the electric potential applied to the actuator to compensate for the mechanical stress __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 1, pp. 79–89, January 2008.     

铁电陶瓷宏观单轴力电行为的双面模型

铁电陶瓷以其优越的力电耦合性作为新型的智能材料使用. 提出基于弹塑性双面理论的宏观铁电本构模型. 根据铁电陶瓷内部电畴在外电场和机械场作用下的微观运动，在宏观上除引入材料的畴变面外，还首次引入饱和面，并考虑以畴变面与饱和面之间的广义距离来表征铁电陶瓷的非线性行为. 数值计算结果与实验数据的比较表明所提出的初步理论可适当地反映力电加载下铁电陶瓷的宏观非线性行为.     

Crack in functionally graded piezoelectric strip bonded to elastic surface layers under electromechanical loading

Solved is the problem of a crack in a functionally graded piezoelectric material (FGPM) bonded to two elastic surface layers. It is assumed that the elastic stiffness, piezoelectric constant, and dielectric permittivity of the FGPM vary continuously along the thickness of the strip. The outside layers are under antiplane mechanical loading and in-plane electric loading. The solution involves solving singular integral equations by application of the Gauss–Jacobi integration formula. Numerical calculations are carried out to obtain the energy density factors. Their variations with the geometric, loading and material parameters are shown graphically.     

Crack growth under alternating loading

Crack propagation under alternating loading is investigated. Relations between the growth rate of a fatigue defect and loading parameters and the expression for the stress intensity factor are derived for compression of a cracked solid taking into account the possible contact of the crack faces. A model for the deformation of a small region near the crack tip is proposed which allows one to formulate the conditions of residual opening of a growing fatigue crack. The experimental data obtained in tests of steel samples are compared with the results of calculation using the developed procedure. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 190–198, September–October, 2008.     

厚壁圆筒的载荷特征与变形规律研究

为了研究厚壁圆筒在受到相同冲量但载荷特征不同时材料的变形规律,分别采用解析解、LS-DYNA软件和有限元计算程序对厚壁圆筒在受到内压作用下的力学响应进行了计算比较,在得到基本一致的结果后,利用有限元计算程序对冲量相同时四种加载方式下厚壁圆筒的动力学响应进行了计算,计算结果表明冲量相同时,不同载荷特征的载荷作用下的应变是不同的,应变与载荷作用时间和大小相关.     

Singular electro-mechanical fields near the apex of a piezoelectric bonded wedge under antiplane shear

This paper presents the explicit forms of singular electro-mechanical field in a piezoelectric bonded wedge subjected to antiplane shear loads. Based on the complex potential function associated with eigenfunction expansion method, the eigenvalue equations are derived analytically. Contrary to the anisotropic elastic bonded wedge, the results of this problem show that the singularity orders are single-root and may be complex. The stress intensity factors of electrical and mechanical fields are dependent. However, when the wedge angles are equal (α=β), the orders become real and double-root. The real stress intensity factors of electrical and mechanical fields are then independent. The angular functions have been validated when they are compared with the results of several degenerated cases in open literatures.     

Elastodynamic Green's functions for a laminated piezoelectric cylinder

Elastodynamic Green's functions for a piezoelectric structure represent the electro-mechanical response due to a steady-state point source as either a unit force or a unit charge. Herein, Green's functions for a laminated circular piezoelectric cylinder are constructed by means of the superposition of modal data from the spectral decomposition of the operator of the equations governing its dynamic behavior. These governing equations are based on a semi-analytical finite element formulation where the discretization occurs through the cylinder's thickness. Examples of a homogeneous PZT-4 cylinder and a two-layer cylinder composed of a PZT-4 material at crystal orientations of ±30° with the longitudinal axis are presented. Numerical implementation details for these two circular cylinders show the convergence and accuracy of these Green's functions.     

Poromechanical response of a finite elastic cylinder under axisymmetric loading

Drained or undrained cylindrical specimens under axisymmetric loading are commonly used in laboratory testing of soils and rocks. Poroelastic cylindrical elements are also encountered in applications related to bioengineering and advanced materials. This paper presents an analytical solution for an axisymmetrically-loaded solid poroelastic cylinder of finite length with permeable (drained) or impermeable (undrained) hydraulic boundary conditions. The general solutions are derived by first applying Laplace transforms with respect to the time and then solving the resulting governing equations in terms of Fourier–Bessel series, which involve trigonometric and hyperbolic functions with respect to the z-coordinate and Bessel functions with respect to the r-coordinate. Several time-dependent boundary-value problems are solved to demonstrate the application of the general solution to practical situations. Accuracy of the numerical solution is confirmed by comparing with the existing solutions for the limiting cases of a finite elastic cylinder and a poroelastic cylinder under plane strain conditions. Selected numerical results are presented for different cylinder aspect ratios, loading and hydraulic boundary conditions to demonstrate the key features of the coupled poroelastic response.     

Nonstationary radial electroelastic vibrations of a piezoceramic cylinder under mechanical loading

The radial nonstationary vibrations of a piezoceramic cylinder polarized across the thickness and subjected to mechanical dynamic loading are studied. To solve the initial–boundary-value problem, mesh approximations and difference schemes are used. The electromechanical state of the cylinder under an instantaneously applied constant force is analyzed in detail     

Crack kinking in a piezoelectric solid

The intrinsic coupling between the mechanical and the electric fields assigns a uniquefeature for the fracture in a piezoelectric solid. We model the kink of a crack by continuousdistribution of edge dislocations and electric dipoles. The problem admits an approach based onthe Stroh formalism. A set of coupled singular integral equations are derived for the dislocationand electric dipole density functions associated with a kinked crack. Numerical results indicatethat the crack tends to propagate in a straight line under a tensile stress and a positive electricfield. For a crack subjected to the mixed mode mechanical loading, a superimposed positiveelectric field tends to reduce the kink angle. The influence of the non-singular T-stress-chargeparallel to a crack is also investigated. It is shown that a transverse tensile stress or a positivetransverse electric field will lead to further deviation of the kinked crack from the crackextension line.     

Generalized thermoelastic analysis of the stress-focusing effect in a solid cylinder under rotationally asymmetrical instantaneous thermal loading

Summary  This paper investigates the stress-focusing effect in an infinitely long cylinder under rotationally asymmetrical instantaneous thermal loading on the basis of the generalized thermoelastic Lord–Shulman (L-S) and Green–Lindsay (G-L) theories. Combined forms of the governing equations of both theories are given in a cylindrical coordinate system. The two-dimensional generalized thermoelastic problems are solved by numerical inversion of Laplace transform. Calculations have been performed to find distributions of thermal stresses on the basis of the L-S theory. Stress-focusing phenomena under different heating conditions are presented. The effects of thermomechanical coupling and relaxation time on the stress-focusing phenomena as well as the singularity of stresses are discussed. Received 15 November 2000; accepted for publication 15 November 2001     

Crack nucleation in a strip of varying thickness under force loading

A mathematical model of crack nucleation in a strip of varying thickness under force loading is constructed. It is assumed that the loading of the strip by external forces gives rise to prefracture zones, which are modeled as regions of weakened interparticle bonds in the material. Solution of the problem of the equilibrium of an isotropic strip of varying thickness, with a crack nucleus reduces to solving a system of nonlinear singular integral equations with a Cauchy type kernel to determine the forces in the region of crack nucleation. The condition of crack nucleation in a strip of varying thickness is formulated using the criterion of the limiting stretching of material bonds.     

Crack growth in unidirectional graphite-epoxy under biaxial loading

Center-notched coupons of 16-ply unidirectional AS4/3501-6 graphite-epoxy were loaded in tension to failure in order to determine the direction of initial crack growth, the load required to cause initial crack growth, and the load required to cause failure. Fifteen-degree off-axis tensile coupons provided a far-field biaxial state of stress away from the notch. Specimens of three different length-to-width ratios were tested in order to vary the biaxial stress states. Two notch configurations were used, one perpendicular to the loading direction, the other perpendicular to the material fiber direction. In all of the cases studied experimentally, crack growth was parallel to the fiber direction. An anisotropicelasticity crack-tip stress analysis was subsequently implemented in order to compare experimental crack-growth directions to those predicted by three crack-extension-direction criteria. Of the three criteria studied, (Strain-energy density, the tensor polynomial, and the normal stress ratio), only one, the normal-stress-ratio criterion, provided crack-growth-direction predictions that agreed with the experimental results. Paper was presented at the 1985 SEM Spring Conference on Experimental Mechanics, held in Las Vegas, NV on June 9–14.