Electroelastic analysis of a penny-shaped crack in a piezoelectric ceramic under mode I loading

Following the theory of linear piezoelectricity, we consider the electroelastic problem for a piezoelectric ceramic with a penny-shaped crack under mode I loading. The problem is formulated by means of Hankel transform and the solution is solved exactly. The stress intensity factor, energy release rate and energy density factor for the exact and impermeable crack models are expressed in closed form and compared for a P-7 piezoelectric ceramic. Based on current findings, we suggest that the energy release rate and energy density factor criteria for the exact crack model are superior to fracture criteria for the impermeable crack model.     

The lowest order solution of the crack–inhomogeneity interaction for mode I crack

The lowest order solution to crack–inhomogeneity interaction is derived for mode I crack. The basic solution evaluates the variation of near-tip stress intensity factors induced by an inhomogeneity of arbitrary shape. A set of simplified forms of the basic solution is also obtained for several special inhomogeneity shapes. As validated by numerical examples, the approximate solutions have good accuracy.     

Stress state of a piezoelectric ceramic body with a plane crack under antisymmetric loads

The static equilibrium of an electroelastic transversely isotropic space with a plane crack under antisymmetric mechanical loads is studied. The crack is located in the plane of isotropy. Relationships are established between the stress intensity factors (SIFs) for an infinite piezoceramic body and the SIFs for a purely elastic body with a crack of the same form under the same loads. This makes it possible to find the SIFs for an electroelastic body without the need to solve specific electroelasitc problems. As an example, the SIFs are determined for a piezoelastic body with penny-shaped and elliptic cracks under shear __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 2, pp. 32–42, February 2006.     

Penny-shaped crack in transversely isotropic piezoelectric materials

Using a method of potential functions introduced successively to integrate the field equations of three-dimensional problems for transversely isotropic piezoelectric materials, we obtain the so-called general solution in which the displacement components and electric potential functions are represented by a singular function satisfying some special partial differential equations of 6th order. In order to analyse the mechanical-electric coupling behaviour of penny-shaped crack for above materials, another form of the general solution is obtained under cylindrical coordinate system by introducing three quasi-harmonic functions into the general equations obtained above. It is shown that both the two forms of the general solutions are complete. Furthermore, the mechanical-electric coupling behaviour of penny-shaped crack in transversely isotropic piezoelectric media is analysed under axisymmetric tensile loading case, and the crack-tip stress field and electric displacement field are obtained. The results show that the stress and the electric displacement components near the crack tip have (r ^−1/2) singularity. The project supported by the Natural Science Foundation of Shaanxi Province, China     

On the stress state of a piezoceramic body with a flat crack under symmetric loads

A static-equilibrium problem is solved for an electroelastic transversely isotropic medium with a flat crack of arbitrary shape located in the plane of isotropy. The medium is subjected to symmetric mechanical and electric loads. A relationship is established between the stress intensity factor (SIF) and electric-displacement intensity factor (EDIF) for an infinite piezoceramic body and the SIF for a purely elastic material with a crack of the same shape. This allows us to find the SIF and EDIF for an electroelastic material directly from the corresponding elastic problem, not solving electroelastic problems. As an example, the SIF and EDIF are determined for an elliptical crack in a piezoceramic body assuming linear behavior of the stresses and the normal electric displacement on the crack surface __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 11, pp. 67–77, November 2005.     

Dynamic response of a crack in a functionally graded interface of two dissimilar piezoelectric half-planes

Summary  In this paper, the dynamic anti-plane crack problem of two dissimilar homogeneous piezoelectric materials bonded through a functionally graded interfacial region is considered. Integral transforms are employed to reduce the problem to Cauchy singular integral equations. Numerical results illustrate the effect of the loading combination parameter λ, material property distribution and crack configuration on the dynamic stress and electric displacement intensity factors. It is found that the presence of the dynamic electric field could impede of enhance the crack propagation depending on the time elapsed and the direction of applied electric impact. Received 4 December 2001; accepted for publication 9 July 2002 This work is supported by the National Natural Science Foundation of China through Grant No. 10132010.     

Thermostressed state of a piezoelectric bodywith a plane crack under symmetric thermal load

The paper addresses a thermoelectroelastic problem for a piezoelectric body with an arbitrarily shaped plane crack in a plane perpendicular to the polarization axis under a symmetric thermal load. A relationship between the intensity factors for stress (SIF) and electric displacement (EDIF) in an infinite piezoceramic body with a crack under a thermal load and the SIF for a purely elastic body with a crack of the same shape under a mechanical load is established. This makes it possible to find the SIF and EDIF for an electroelastic material from the elastic solution without the need to solve specific problems of thermoelasticity. The SIF and EDIF for a piezoceramic body with an elliptic crack and linear distribution of temperature over the crack surface are found as an example __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 3, pp. 96–108, March 2008.     

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     

Point forces and point charge applied to a circular crack in a transversely isotropic piezoelectric space

Interaction between an arbitrarily located and oriented point force and point charge with a circular crack is considered. Obtained are the exact expressions for the stress intensity factors (SIFs) k_j (j=1,2,3) and electric displacement intensity factor (EDIF) k_D; they are given in terms of elementary functions. The results are also presented in graphical form.     

Penny-shaped and half-plane cracks in a transversely isotropic piezoelectric solid under arbitrary loading

Summary The problem of a penny-shaped crack in a transversely isotropic piezoelectric material loaded by both normal and tangential tractions and by electric charges is analyzed. Closed-form solutions are obtained for the full electroelastic fields as well as for the stress and electric displacement intensity factors. Solutions are also obtained for the (non-trivial) limiting case of a half-plane crack. The results are illustrated on the example of piezoceramics PZT-6B. Received 12 July 1999; accepted for publication 20 July 1999     

A moving mode-III crack in functionally graded piezoelectric material: permeable problem

In this paper a moving mode-III crack in functionally graded piezoelectric materials (FGPM) is studied. The crack surfaces are assumed to be permeable. The governing equations for FGPM are solved by means of Fourier cosine transform. The mathematical formulation for the permeable crack condition is derived as a set of dual integral equations, which, in turn, are reduced to a Fredholm integral equation of the second kind. The results obtained indicate that the stress intensity factor of moving crack in FGPM depends only on the mechanical loading. The gradient parameter of the FGPM and the moving velocity of the crack do have significant influence on the dynamic stress intensity factor.     

A penny-shaped crack in a transversely isotropic piezoelectric layer

In this paper, we develop a model to treat penny-shaped crack configuration in a piezoelectric layer of finite thickness. The piezoelectric layer is subjected to axially symmetric mechanical and electrical loads. Hankel transform technique is used to reduce the problem to the solution of a system of integral equations. A numerical solution for the crack tip fields is obtained for different crack radius and crack position.     

Exact solution of a semi-infinite crack in an infinite piezoelectric body

Summary The paper presents an exact and complete solution of the problem of a semi-infinite plane crack in an infinite transversely isotropic piezoelectric body. The upper and lower crack faces are assumed to be loaded symmetrically by a couple of normal point forces in opposite directions and a couple of point charges. The solution is derived through a limiting procedure from the one of a penny-shaped crack. The expressions for the elastoelectric field are given in terms of elementary functions. Received 10 August 1998; accepted for publication 18 November 1998     

Impact response of an interface crack in a hybrid piezoelectric laminate

Summary  In a hybrid laminate containing an interfacial crack between piezoelectric and orthotropic layers, the dynamic field intensity factors and energy release rates are obtained for electro-mechanical impact loading. The analysis is performed within the framework of linear piezoelectricity. By using integral transform techniques, the problem is reduced to the solution of a Fredholm integral equation of the second kind, which is obtained from one pair of dual integral equations. Numerical results for the dynamic stress intensity factor show the influence of the geometry and electric field. Received 29 June 2001; accepted for publication 3 December 2001     

残余应力下厚壁筒表面裂纹的应力强度因子计算

本文首先介绍了边界元法计算裂纹尖端应力强度因子的基本理论,接着利用边界元法计算了在残余应力下不同厚壁筒内表面椭圆裂纹的应力强度因子,研究了其大不随椭圆裂纹不同而变化的规律,为厚壁筒结构的设计,制造以及疲劳寿命分析提供了许多有价值的参考资料。     

A penny-shaped interface crack between a functionally graded piezoelectric layer and a homogeneous piezoelectric layer

The problem of a penny-shaped interface crack between a functionally graded piezoelectric layer and a homogeneous piezoelectric layer is investigated. The surfaces of the composite structure are subjected to both mechanical and electrical loads. The crack surfaces are assumed to be electrically impermeable. Integral transform method is employed to reduce the problem to a Fredholm integral equation of the second kind. The stress intensity factor, electric displacement intensity factor and energy release rate are derived, some typical numerical results are plotted graphically. The effects of electrical loads, material nonhomogeneity and crack configuration on the fracture behaviors of the cracked composite structure are analyzed in detail.     

Dislocation emission criterion from a blunt crack tip

Simple closed-form solutions for critical stress intensity factors were developed for edge dislocation emission from a blunt crack under mode I and mode II loading conditions. The solutions are applicable both for a slender elliptical hole and a long slit with blunted head. When the blunt crack considered degenerates to a sharp crack, the present solutions reduce to those previously reported in literature.     

Elastic state of a transversely isotropic piezoelectric body with an arbitrarily oriented elliptic crack

The elastic stress state in a piezoelectric body with an arbitrarily oriented elliptic crack under mechanical and electric loads is analyzed. The solution is obtained using triple Fourier transform and the Fourier-transformed Green’s function for an unbounded piezoelastic body. Solving the problem for the case of a crack lying in the isotropy plane, for which there is an exact solution, demonstrates that the approach is highly efficient. The distribution of the stress intensity factors along the front of a crack in a piezoelectric body under uniform mechanical loading is analyzed numerically for different orientations of the crack __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 39–48, February 2008.     

Stress state of a transversely isotropic magnetoelectroelastic body with a plane crack under antisymmetric loads

The static equilibrium of a transversely isotropic magnetoelectroelastic body with a plane crack of arbitrary shape in the isotropy plane under antisymmetric mechanical loading is studied. The relationships between the stress intensity factors (SIFs) for an infinite magnetoelectroelastic body and the SIFs for a purely elastic body with the same crack and under the same antisymmetric loading are established. This enables the SIFs for a magnetoelectroelastic body to be found directly from the analogous problem of elasticity. As an example of using this result, the SIFs for penny-shaped, elliptic, and parabolic cracks in a magnetoelectroelastic body under antisymmetric mechanical loading are found Translated from Prikladnaya Mekhanika, Vol. 44, No. 10, pp. 37–51, October 2008.     

Electro-mechanical analysis of an interfacial crack between a piezoelectric and two orthotropic layers

Summary  The problem of an interfacially cracked three-layered structure constructed of a piezoelectric and two orthotropic materials is analyzed using the theory of linear piezoelectricity and fracture mechanics. Anti-plane shear loading is considered, and the integral transform technique is used to determine the stress intensity factor. Numerical examples show the electro-mechanical effects of various material combinations and layer thicknesses on the stress intensity factor. Interesting results are obtained in comparison with earlier solutions for interfacially cracked piezoelectric structures. Received 29 December 2000; accepted for publication 3 May 2001