Anti-plane shear of an arbitrary oriented crack in a functionally graded strip bonded with two dissimilar half-planes

An internal crack located within a functionally graded material (FGM) strip bonded with two dissimilar half-planes and under an anti-plane load is considered. The crack is oriented in an arbitrary direction. The material properties of strip are assumed to vary exponentially in the thickness direction and two half-planes are assumed to be isotropic. Governing differential equations are derived and to reduce the difficulty of the problem dealing with solution of a system of singular integral equations Fourier integral transform is employed. Semi closed form solution for the stress distribution in the medium is obtained and mode III stress intensity factor (SIF), at the crack tip is calculated and its validity was verified. Finally, the effects of nonhomogeneous material parameter and crack orientation on the stress intensity factor are studied.     

Mode-III interface edge crack between two bonded quarter-planes of dissimilar piezoelectric materials

Summary  The problem of an interface edge crack between two bonded quarter-planes of dissimilar piezoelectric materials is considered under the conditions of anti-plane shear and in-plane electric loading. The crack surfaces are assumed to be impermeable to the electric field. An integral transform technique is employed to reduce the problem under consideration to dual integral equations. By solving the resulting dual integral equations, the intensity factors of the stress and the electric displacement and the energy release rate as well as the crack sliding displacement and the electric voltage across the crack surfaces are obtained in explicit form for the case of concentrated forces and free charges at the crack surfaces and at the boundary. The derived results can be taken as fundamental solutions which can be superposed to model more realistic problems. Received 10 November 2000; accepted for publication 28 March 2001     

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     

Propagation of an anti-plane moving crack in a functionally graded piezoelectric strip

Summary The propagation of an anti-plane moving crack in a functionally graded piezoelectric strip (FGPS) is studied in this paper. The governing equations for the proposed analysis are solved using Fourier cosine transform. The mixed boundary value problems of the anti-plane moving crack, which is assumed to be either impermeable or permeable, are formulated as dual integral equations. By appropriate transformations, the dual integral equations are reduced to Fredholm integral equations of the second kind. For the impermeable crack, the stress intensity factor (SIF) of the crack in the FGPS depends on both the mechanical and electric loading, whereas, the SIF for the permeable crack depends only on the mechanical loading. The results obtained show that the gradient parameter of the FGPS and the velocity of the crack have significant influence on the dynamic SIF.Support from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7081/00E) is acknowledged. Support from the National Natural Science Foundation of China (Project No. 10072041) is also acknowledged.     

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.     

An anti-plane propagating crack in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric strip

Summary A finite crack propagating at constant speed in a functionally graded piezoelectric strip (FGPS) bonded to a homogeneous piezoelectric strip is considered. It is assumed that the electroelastic material properties of the FGPS vary exponentially across the thickness of the strip, and that the bimaterial strip is under combined anti-plane mechanical shear and in-plane electrical loads. The analysis is conducted for the electrically unified crack boundary condition, which includes both the traditional permeable and the impermeable ones. By using the Fourier transform, the problem is reduced to the solution of Fredholm integral equations of the second kind. Numerical results for the stress intensity factor and the crack sliding displacement are presented to show the influences of the crack propagation speed, electric loads, FGPS gradation, crack length, electromechanical coupling coefficient, properties of the bonded homogeneous piezoelectric strip and crack location.     

Anti-plane problem of periodic cracks in a functionally graded coating–substrate structure

The static and dynamic anti-plane problem for a functionally graded coating–substrate structure containing a periodic array of parallel cracks, which are perpendicular to the boundary, is considered. Integral-transform techniques are employed to reduce the problem to the solution of an integral equation with hypersingular kernels. Numerical results are presented to show the influence of geometry, material properties and material gradient parameter on the fracture behavior.     

Torsional impact of transversely isotropic solid with functionally graded shear moduli and a penny-shaped crack

The torsional impact response of a penny-shaped crack in an unbounded transversely isotropic solid is considered. The shear moduli are assumed to be functionally graded such that the mathematics is tractable. Laplace transform and Hankel transform are used to reduce the problem to solving a Fredholm integral equation. The crack tip stress fields are obtained. Investigated are the influence of material nonhomogeneity and orthotropy on the dynamic stress intensity factor. The peak value of the dynamic stress intensity factor can be suppressed by increasing the shear moduli's gradient and/or increasing the shear modulus in a direction perpendicular to the crack surface.     

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.     

Analytical solution for the electromechanical behavior of a rotating functionally graded piezoelectric hollow shaft

In the present paper, the analytical solution for a radially piezoelectric functionally graded rotating hollow shaft is presented. The variation of material properties is assumed to follow a power law along the radial direction of the shaft. Two resulting fully coupled differential equations in terms of the displacement and electric potential are solved directly. Numerical results for different shaft geometries with different profiles of inhomogeneity are also graphically displayed.     

Transient response of an insulating crack between dissimilar piezoelectric layers under mechanical and electrical impacts

Summary  The dynamic response of an interface crack between two dissimilar piezoelectric layers subjected to mechanical and electrical impacts is investigated under the boundary condition of electrical insulation on the crack surface by using the integral transform and the Cauchy singular integral equation methods. The dynamic stress intensity factors, the dynamic electrical displacement intensity factor, and the dynamic energy release rate (DERR) are determined. The numerical calculation of the mode-I plane problem indicates that the DERR is more liable to be the token of the crack growth when an electrical load is applied. The dynamic response shows a significant dependence on the loading mode, the material combination parameters as well as the crack configuration. Under a given loading mode and a specified crack configuration, the DERR of an interface crack between piezoelectric media may be decreased or increased by adjusting the material combination parameters. It is also found that the intrinsic mechanical-electrical coupling plays a more significant role in the dynamic fracture response of in-plane problems than that in anti-plane problems. Received 4 September 2001; accepted for publication 23 July 2002 The work was supported by the National Natural Science Foundation under Grant Number 19891180, the Fundamental Research Foundation of Tsinghua University, and the Education Ministry of China.     

Effects of the weak/micro-discontinuity of interface on the fracture behavior of a functionally graded coating with an inclined crack

The mechanical model was established for the anti-plane fracture problem of a functionally graded coating–substrate system with a coating crack inclined to the weak/micro-discontinuous interface. The Cauchy singular integral equation for the crack was derived using Fourier integral transform, and the Lobatto–Chebyshev collocation method put up by Erdogan and Gupta was used to get its numerical solution. Finally, the effects of the weak/micro-discontinuity of the interface on SIFs were analyzed, the “affected regions” corresponding to the two crack tips have been obtained and their engineering significance was discussed. It was indicated that, for the crack tip in the corresponding “affected region”, to reduce the weak-discontinuity of the interface and to make the interface micro-discontinuous are the two effective ways to reduce the SIF, and the latter way always has more remarkable SIF-reduction effect. For the crack tip outside the “affected region”, its SIF is mainly influenced by material stiffness, and to prevent such a tip from growing toward the interface “softer coating and stiffer substrate” is a more advantageous combination than “stiffer coating and softer substrate”.     

Impact response of a functionally graded piezoelectric plate with a vertical crack

The dynamic fracture problem for a functionally graded piezoelectric plate containing a crack perpendicular to the free boundaries is considered in this study. It is assumed that the electroelastic properties of the medium vary continuously in the thickness direction. Integral transform techniques and dislocation density function are employed to reduce the problem to the solution of a singular integral equation. Mode I dynamic energy density factors are presented for an internal crack as well as an edge crack for various values of dimensionless parameters representing the size and location of the crack and the material nonhomogeneity.     

The interaction of two parallel Mode-I limited-permeable cracks in a functionally graded piezoelectric material

In this paper, the interaction of two parallel Mode-I limited-permeable cracks in a functionally graded piezoelectric material was investigated by using the generalized Almansi's theorem. In the analysis, the electric permittivity of the air inside the crack was considered. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which unknown variables are jumps of displacements across the crack surface. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. The solution of the present paper shows that the singular stresses and the singular electric displacements at the crack tips in functionally graded piezoelectric materials carry the same forms as those in homogeneous piezoelectric materials; however, the magnitudes of intensity factors depend on the electric permittivity of the air inside the crack and the gradient parameter of functionally graded piezoelectric material properties. It was also revealed that the crack shielding effect is also present in functionally graded piezoelectric materials.     

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     

Anti-plane stress intensity, energy release and energy density at crack tips in a functionally graded strip with linearly varying properties

The fracture problem of a crack in a functionally graded strip with its properties varying in a linear form along the strip thickness under an anti-plane load is considered. The embedded anti-plane crack is located in the middle of strip half way through the thickness. The third mode stress intensity factor is derived using two different methods. In the first method, by employing Fourier integral transforms, the governing equation is converted to a singular integral equation, which is subsequently solved numerically by the collocation method based on Chebyshev polynomials. Then, the problem is solved by means of finite element method in which quadrilateral 8-node singular elements around each crack tip are used. After inspecting the validity of the solution technique, effects of crack geometry and non-homogeneous material parameter on the stress intensity, energy release and energy density are studied and the results of analytical and FEM solutions are compared.     

Analysis of cracked functionally graded piezoelectric strip

The fracture behavior of a cracked strip under antiplane mechanical and inplane electrical loading is studied. A functionally graded piezoelectric strip with exponential material gradation is under consideration. The mechanical and electrical loading is combined via loading coupling factor. The problem of a graded piezoelectric strip containing a screw dislocation is solved. This solution results in stress and electric displacement components with Cauchy singularity. Based on the solution achieved for the dislocation, the distributed dislocation technique (DDT) is utilized to form any geometry of multiple cracks and analyze the behavior of a cracked strip under antiplane mechanical and inplane electrical loading. This technique is capable of the analysis of a strip with a system of interacting cracks. Several examples including strips with single crack, two straight cracks and two curved cracks are presented.     

The behavior of a crack in functionally graded piezoelectric/piezomagnetic materials under anti-plane shear loading

Summary In this paper, the behavior of a crack in functionally graded piezoelectric/piezomagnetic materials subjected to an anti-plane shear loading is investigated. To make the analysis tractable, it is assumed that the material properties vary exponentially with the coordinate parallel to the crack. By using a Fourier transform, the problem can be solved with the help of a pair of dual integral equations in which the unknown variable is the jump of the displacements across the crack surfaces. These equations are solved using the Schmidt method. The relations among the electric displacement, the magnetic flux and the stress field near the crack tips are obtained. Numerical examples are provided to show the effect of the functionally graded parameter on the stress intensity factors of the crack.The authors are grateful for financial support from the Natural Science Foundation of Hei Long Jiang Province (A0301), the National Natural Science Foundation of China (50232030, 10172030), the Natural Science Foundation with Excellent Young Investigators of Hei Long Jiang Province(JC04-08) and the National Science Foundation with Excellent Young Investigators (10325208).     

The dynamic response of an edge crack in a functionally graded orthotropic strip

The dynamic response of a functionally graded orthotropic strip with an edge crack perpendicular to the boundaries is studied. The material properties are assumed to vary continuously along the thickness direction. Laplace and Fourier transforms are applied to reduce the problem to a singular integral equation. Numerical results are presented to illustrate the influences of parameters such as the nonhomogeneity constant and geometry parameters on the dynamic stress intensity factors (SIFs).     

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.