压电材料中的Bueckner功共轭积分及和J积分M积分的关系

研究横观各向同性压电材料中裂纹问题,提出了Bueckner功共轭积分在这类材料中的表达式：并通过引出两类辅助的应力-位移-电位移-电势场,证明功共轭积分和这类材料中的J积分和M积分仍然存在简单的两倍关系由此,各类在脆性材料断裂问题中已广泛应用的权函数方法可顺理成章地推广到压电材料的研究中来．这对独立地确定电位移强度因子和经典的I、II型应力强度因子提供了有力的数学上的工具．进而通过计算机械应变能释放率对压电材料中裂纹的稳定做出判断．     

压电材料中三维I型断裂力学分析

讨论了不可导通情况下三维横观各向刚性压电材料中受拉伸和电载荷作用的平片裂纹Ⅰ型断裂力学问题．使用自限部分概念，从二维线性压电理论出发，严格得到了一组以裂纹面位移间断和电势间断为未知变量的超奇异积分方程组；应用二维超奇异积分的主部分析法，从理论上分析得到了裂纹前沿应力和电势奇性指数以及应力和电位移奇性场，从而找到了以裂纹面位移间断和电势间断表示的应力和电位移强度因子、能量释放率表达式；为所得到的超奇异积分方程组建立了数值法，并用此计算了若干典型的平片裂纹问题，数值结果令人满意．     

The weight function for cracks in piezoelectrics

The weight function in fracture mechanics is the stress intensity factor at the tip of a crack in an elastic material due to a point load at an arbitrary location in the body containing the crack. For a piezoelectric material, this definition is extended to include the effect of point charges and the presence of an electric displacement intensity factor at the tip of the crack. Thus, the weight function permits the calculation of the crack tip intensity factors for an arbitrary distribution of applied loads and imposed electric charges. In this paper, the weight function for calculating the stress and electric displacement intensity factors for cracks in piezoelectric materials is formulated from Maxwell relationships among the energy release rate, the physical displacements and the electric potential as dependent variables and the applied loads and electric charges as independent variables. These Maxwell relationships arise as a result of an electric enthalpy for the body that can be formulated in terms of the applied loads and imposed electric charges. An electric enthalpy for a body containing an electrically impermeable crack can then be stated that accounts for the presence of loads and charges for a problem that has been solved previously plus the loads and charges associated with an unsolved problem for which the stress and electric displacement intensity factors are to be found. Differentiation of the electric enthalpy twice with respect to the applied loads (or imposed charges) and with respect to the crack length gives rise to Maxwell relationships for the derivative of the crack tip energy release rate with respect to the applied loads (or imposed charges) of the unsolved problem equal to the derivative of the physical displacements (or the electric potential) of the solved problem with respect to the crack length. The Irwin relationship for the crack tip energy release rate in terms of the crack tip intensity factors then allows the intensity factors for the unsolved problem to be formulated, thereby giving the desired weight function. The results are used to derive the weight function for an electrically impermeable Griffith crack in an infinite piezoelectric body, thereby giving the stress intensity factors and the electric displacement intensity factor due to a point load and a point charge anywhere in an infinite piezoelectric body. The use of the weight function to compute the electric displacement factor for an electrically permeable crack is then presented. Explicit results based on a previous analysis are given for a Griffith crack in an infinite body of PZT-5H poled orthogonally to the crack surfaces.     

刚度微分法计算压电材料平面断裂问题

把计算应变能释放率的刚度微分法推广到压电材料平面断裂问题．在此基础上，利用压电材料平面断裂问题的有限元数值解作为真实场，用Ｓｏｓａ的平面问题裂端渐近解作为辅助场，由推广的交互Ｍ积分法求得了应力强度因子ＫＩ，ＫＩＩ和电位移强度因子ＫＩＶ．算例表明，计算结果与理论解符合得很好     

Numerical solution of polarization saturation/dielectric breakdown model in 2D finite piezoelectric media

The polarization saturation (PS) model [Gao, H., Barnett, D.M., 1996. An invariance property of local energy release rates in a strip saturation model of piezoelectric fracture. Int. J. Fract. 79, R25–R29; Gao, H., Zhang, T.Y., Tong, P., 1997. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J. Mech. Phys. Solids 45, 491–510], and the dielectric breakdown (DB) model [Zhang, T.Y., Zhao, M.H., Cao, C.F., 2005. The strip dielectric breakdown model. Int. J. Fract. 132, 311–327] explain very well some experimental observations of fracture of piezoelectric ceramics. In this paper, the nonlinear hybrid extended displacement discontinuity-fundamental solution method (NLHEDD-FSM) is presented for numerical analysis of both the PS and DB models of two-dimensional (2D) finite piezoelectric media under impermeable and semi-permeable electric boundary conditions. In this NLHEDD-FSM, the solution is expressed approximately by a linear combination of fundamental solutions of the governing equations, which includes the extended point force fundamental solutions with sources placed at chosen points outside the domain of the problem under consideration, and the extended Crouch fundamental solutions with extended displacement discontinuities placed on the crack and the electric yielding zone. The coefficients of the fundamental solutions are determined by letting the approximated solution satisfy certain conditions on the boundary of the domain, on the crack face and the electric yielding zone. The zero electric displacement intensity factor in the PS model or the zero electric field strength intensity factor in the DB model at the outer tips of the electric yielding zone is used as a supplementary condition to determine the size of the electric yielding zone. Iteration approaches are adopted in the NLHEDD-FSM. The electric yielding zone is determined, and the extended intensity factors and the local J-integral are calculated for center cracks in piezoelectric strips. The effects of finite domain size, saturation property and different electric boundary conditions, as well as different models on the electric yielding zone and the local J-integral, are studied.     

CONSERVATION LAWS IN FINITE MICROCRACKING BRITTLE SOLIDS

This paper addresses the conservation laws in finite brittle solids with microcracks. The discussion is limited to the 2-D cases. First, after considering the combination of the Pseudo-Traction Method and the indirect Boundary Element Method, a versatile method for solving multi-crack interacting problems in finite plane solids is proposed, by which the fracture parameters （SIF and path-independent integrals） can be calculated with a desirable accuracy. Second, with the aid of the method proposed, the roles the conservation laws play in the fracture analysis for finite microcracking solids are studied. It is concluded that the conservation laws do play important roles in not only the fracture analysis but also the analysis of damage and stability for the finite microcracking system. Finally, the physical interpretation of the M-integral is discussed further. An explicit relation between the M-integral and the crack face area, i.e., M = GS, has been discovered using the analytical method, which can shed some light on the Damage Mechanics issues from a different perspective.     

Invariant formulation of the electromechanical enthalpy function of transversely isotropic piezoelectric materials

Summary Theoretical and numerical aspects of the formulation of electromechanically coupled, transversely isotropic solids are discussed within the framework of the invariant theory. The main goal is the representation of the governing constitutive equations for reversible material behaviour based on an anisotropic electromechanical enthalpy function, which automatically fulfills the requirements of material symmetry. The introduction of a preferred direction in the argument list of the enthalpy function allows the construction of isotropic tensor functions, which reflect the inherent geometrical and physical symmetries of the polarized medium. After presenting the general framework, we consider two important model problems within this setting: i) the linear piezoelectric solid; and ii) the nonlinear electrostriction. A parameter identification of the invariant- and the common coordinate-dependent formulation is performed for both cases. The tensor generators for the stresses, electric displacements and the moduli are derived in detail, and some representative numerical examples are presented.We thank Dipl.-Ing. H. Romanowski for his support and helpful remarks.     

Dynamic fracture behavior of a cracked piezoelectric half space under anti-plane mechanical and in-plane electric impact

Summary  The dynamic response of a cracked piezoelectric half-space under anti-plane mechanical and in-plane electric impacting loads is investigated in the present paper. In the study, the crack is assumed parallel to the free surface of the half-space. Laplace and Fourier transforms are used to reduce the mixed boundary value problems to Cauchy-type singular integral equations in the Laplace transform domain, which are solved numerically. Then, a numerical Laplace inversion is performed and the dynamic stress and electric displacement factors are obtained as functions of time and geometry parameters. The dynamic energy release rate is derived for piezoelectric materials in terms of the electroelastic intensities and is displayed graphically. Received 5 January 2000; accepted for publication 28 June 2000     

压电复合材料中正n边形孔边裂纹的反平面问题研究

本文基于Cauchy积分理论和Schwarz-Christoffel(SC)变换技术,针对压电复合材料中带一条裂纹的正n边形孔口缺陷的反平面断裂力学进行了探究.假设满足电不可通边界条件,利用Cauchy积分公式和留数定理,获得了任意正n边形裂尖处应力和电位移两个场强度因子以及全能量释放率的封闭形式的显式解.当正n边形边...     

Thermo-electro-elastic transient responses in piezoelectric hollow structures

The paper presents an analytical method to solve thermo-electro-elastic transient response in piezoelectric hollow structures subjected to arbitrary thermal shock, sudden mechanical load and electric excitation. Volterra integral equation of the second kind caused by interaction between elastic deformation and electric field is solved by using an interpolation method. Thus, the exact expressions for the transient responses of displacement, stresses, electric displacement and electric potential in the piezoelectric hollow structures are obtained by means of Hankel transform, Laplace transform, and their inverse transforms. In Section 2, based on spherical coordinates, the governing equation of thermo-electro-elastic transient responses in a piezoelectric hollow sphere is found and the associated numerical results are carried out. In Section 3, based on cylindrical coordinates, the governing equation of thermo-electro-elastic transient responses in a non-homogeneous piezoelectric hollow cylinder is found and the corresponding numerical results are carried out. The results carried out may be used as a reference to solve other transient coupled problems of thermo-electro-elasticity in piezoelectric structures.     

M积分与夹杂/缺陷弹性模量的显式关系

M积分在材料构型力学中表征着缺陷自相似扩展的能量释放率,而有效弹性模量下降量在传统损伤力学中是一个具有内变量属性的损伤参数. 探讨了两者之间的特定关系,以此为材料构型力学与损伤力学搭建桥梁.借助穆斯海里什维利（Muskhelishvili）复势函数方法获取无限大弹性平面含圆形夹杂的弹性场解,根据M 积分的复势函数解析表达式得到M 积分与夹杂弹性模量的显式表达式. 随后通过有限元分析,对含复杂缺陷群的弹塑性材料进行数值模拟,结果表明内部缺陷区域的有效弹性模量下降与M 积分存在着特定关系. 基于此,提出利用材料构型力学中的外变量参数（M 积分）来替代损伤力学中的内变量参数（弹性模量下降量）描述材料的缺陷演化.      

Measurement of the M-integral for a Hole in an Aluminum Plate or Strip

A method using Digital Image Correlation (DIC) is proposed to measure the M-integral in an elastic rectangular plate and elastic–plastic strip made of LY12 Aluminum where a hole is located at center as a defect. The path-independence property of the M-integral is verified by selecting a few of closed contours to evaluate the M-integral. It is found that the measured values of the M-integral are path-independent when the closed contours are far from the nonlinear plastic zone. In contrast, large deviations occur in determining the M-integral among different integral contours when the contours pass through the plastic zone. The present study demonstrates that DIC method used by the ARAMIS 4 M instrument and the proposed smoothing technique for evaluating the measured displacements do provide the effective tools to measure the M-integral in describing the local damage of elastic and elastic–plastic materials. This technique could be extended to measure the M-integral for other complicated damage, e.g., multiple defects with different shapes in a local region.     

Magneto-electro-elastic antiplane analysis of a bimaterial BaTiO3–CoFe2O4 composite wedge with an interface crack

The antiplane analysis is made for a bimaterial BaTiO₃–CoFe₂O₄ composite wedge containing an interface crack. The coupled magneto-electro-elastic field is induced by the piezoelectric/piezomagnetic BaTiO₃–CoFe₂O₄ composite materials. For the crack problems, the intensity factors of stress, strain, electric displacement, electric field, magnetic induction and magnetic field at crack tips are derived analytically. Also, the energy density criterion is applied to predict the fracture behavior of the interface crack. The numerical results also show that the energy release rate for a crack in a single wedge is negative.     

Plane problems in piezoelectric media with elliptic inclusion

I.IntroductionPiezoelectricmedia,asa"ex\'typeoffullctionalmaterial.arex'idel}'appliedtomanytechnologicalfieldsduetoitselectronlechallicalcouplillgeffect.Defects.likethatofothermaterials.arenotlimitedtocracks.x'oidsandinclusionsillpiezoelectricmaterialsorelements.Yet,stressconcentrationsornoll-ullitbrllldistl-ibutionsofelectricfieldillducedbythosedefectsareoneofthehe}l'filctorswllicllwouldleadpiezoelectricstructurestonon-normalfailure.Therel'ore.itisofgrealimportancetostudythepropertiesofthos…     

Dimensional reduction of a piezoelectric composite rod

In the present paper, a new generalized Timoshenko model is constructed for a composite rod with embedded or attached piezoelectric materials. This model is applicable to composite rods without prescribed electric potential along the lateral surfaces. The Variational-Asymptotic Method (VAM) is applied as a mathematical tool to carry out the dimensional reduction process. The present reduced model captured the effects of dielectric as well as the polarization of the piezoelectric material, which justifies its coupled electromechanical nature. First, the three-dimensional electromechanical enthalpy is asymptotically approximated by VAM using the slenderness of the rod as the small parameter and subsequently an equivalent one-dimensional electromechanical enthalpy is developed. Energy terms, which are asymptotically correct up to the second order are kept in the approximate enthalpy expression. For engineering applications, the approximate enthalpy is then transformed into a generalized Timoshenko model which has the traditional six mechanical degrees of freedom along with an extra one-dimensional electric degree of freedom.     

Anti-plane analysis of semi-infinite crack in piezoelectric strip

Using the complex variable function method and the technique of the conformal mapping, the fracture problem of a semi-infinite crack in a piezoelectric strip is studied under the anti-plane shear stress and the in-plane electric load. The analytic solutions of the field intensity factors and the mechanical strain energy release rate are presented under the assumption that the surface of the crack is electrically impermeable. When the height of the strip tends to infinity, the analytic solutions of an infinitely large piezoelectric solid with a semi-infinite crack are obtained. Moreover, the present results can be reduced to the well-known solutions for a purely elastic material in the absence of the electric loading. In addition, numerical examples are given to show the influences of the loaded crack length, the height of the strip, and the applied mechanical/electric loads on the mechanical strain energy release rate.     

含椭圆形刚性夹杂的压电材料平面问题

应用复变函数的Ｆａｂｅｒ级数展开方法，本文研究了含椭圆形刚性夹杂的压电材料平面问题，给出了问题的封闭解。解签表明，夹杂内的电场强度和电位移为常量。并通过算例分析，讨论了正，逆压电效应在基体孔周处的机电行为。     

Magneto–thermo–electro–elastic transient response in a piezoelectric hollow cylinder subjected to complex loadings

The article presents an analytical solution for magneto–thermo–electro–elastic problems of a piezoelectric hollow cylinder placed in an axial magnetic field subjected to arbitrary thermal shock, mechanical load and transient electric excitation. Using an interpolation method solves the Volterra integral equation of the second kind caused by interaction among magnetic, thermal, electric and mechanical fields, the electric displacement is determined. Thus, the exact expressions for the transient responses of displacement, stresses, electric displacement, electric potential and perturbation of the magnetic field vector in the piezoelectric hollow cylinder are obtained by means of Hankel transforms, Laplace transforms, and inverse Laplace transforms. From sample numerical calculations, it is seen that the present method is suitable for a piezoelectric hollow cylinder subjected to arbitrary thermal shock, mechanical load and transient electric excitation, and the result carried out may be used as a reference to solve other transient coupled problems of magneto–thermo–electro–elasticity.     

M积分与夹杂/缺陷弹性模量的显式关系简

M积分在材料构型力学中表征着缺陷自相似扩展的能量释放率,而有效弹性模量下降量在传统损伤力学中是一个具有内变量属性的损伤参数. 探讨了两者之间的特定关系,以此为材料构型力学与损伤力学搭建桥梁.借助穆斯海里什维利（Muskhelishvili）复势函数方法获取无限大弹性平面含圆形夹杂的弹性场解,根据M 积分的复势函数解析表达式得到M 积分与夹杂弹性模量的显式表达式. 随后通过有限元分析,对含复杂缺陷群的弹塑性材料进行数值模拟,结果表明内部缺陷区域的有效弹性模量下降与M 积分存在着特定关系. 基于此,提出利用材料构型力学中的外变量参数（M 积分）来替代损伤力学中的内变量参数（弹性模量下降量）描述材料的缺陷演化.     

Non-linear longitudinal vibrations of piezoceramics excited by weak electric fields

Typical non-linear effects, e.g. dependence of the resonance frequency on the amplitude, superharmonics in spectra and a non-linear relationship between excitation voltage and vibration amplitude as well as jump phenomena are observed in experiments with piezoceramics excited at resonance by weak electric fields. These non-linear effects can be observed for both the piezoelectric 31- and the 33-effect. In contrast to the well-known non-linear effects exhibited by piezoceramics in the presence of strong electric fields, these effects are not described in detail in the literature.In this paper, we attempt to model these phenomena using an electric enthalpy density to capture the cubic-like effects observed in the experiments. The equations of motion for the system under consideration are derived via the Ritz method using Hamilton's principle. The ‘non-linear’ parameters are identified and the numerical results are compared to those obtained experimentally. The effects described herein may have a significant influence in structures excited close to resonance frequencies via piezoelectric elements.