Dynamic stress intensity factors of two 3D rectangular permeable cracks in a transversely isotropic piezoelectric material under a time‐harmonic elastic P‐wave

The dynamic behavior of two 3D rectangular permeable cracks in a transversely isotropic piezoelectric material is investigated under an incident harmonic stress wave by using the generalized Almansi's theorem and the Schmidt method. The problem is formulated through double Fourier transform into three pairs of dual integral equations with the displacement jumps across the crack surfaces as the unknown variables. To solve the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. Finally, the relations among the dynamic stress field and the dynamic electric displacement filed near the crack edges are obtained, and the effects of the shape of the rectangular crack, the characteristics of the harmonic wave, and the distance between two rectangular cracks on the stress and the electric intensity factors in a piezoelectric composite material are analyzed. Copyright © 2015 John Wiley & Sons, Ltd.     

Non-local theory solution of a mode-I crack in a piezoelectric/piezomagnetic composite material plane

The non-local theory solution of a mode-I permeable crack in a piezoelectric/piezomagnetic composite material plane was given by using the generalized Almansi’s theorem and the Schmidt method in this paper. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Numerical examples were provided to show the effects of the crack length and the lattice parameter on the stress field, the electric displacement field and the magnetic flux field near the crack tips. Unlike the classical elasticity solutions, it is found that no stress, electric displacement and magnetic flux singularities are present at the crack tips in piezoelectric/piezomagnetic composite materials. The non-local elastic solution yields a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion.     

Interaction of parallel dielectric cracks in functionally graded piezoelectric materials

In this paper, the problem of two interacting parallel cracks in functionally graded piezoelectric materials under in-plane electromechanical loads is studied. The formulation is based on using Fourier transforms and modeling the cracks as distributed dislocations, and the resulting singular integral equations are solved with Chebyshev polynomials. A dielectric crack model considering the crack filling effect is adopted to describe the electric boundary conditions along crack surfaces. Numerical simulations are made to show the effect of material gradient, the geometry of interacting cracks, and crack position upon fracture parameters such as stress intensity factors, electric displacement intensity factor, and COD intensity factor. By considering the effect of a dielectric medium inside the crack and crack deformation, the results obtained from the dielectric crack model are always between those from the traditional crack models with physical limitation.     

Investigation of the behavior of a mode-I crack in functionally graded materials by non-local theory

In this paper, the behavior of a mode-I crack in functionally graded materials is investigated by means of the non-local theory. The traditional concepts of the non-local theory are firstly extended to solve the mode-I crack fracture problem in functionally graded materials, in which the shear modulus varies exponentially with coordinate parallel to the crack. Through the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables are jumps of displacements across crack surfaces, not the dislocation density functions or the analysis functions. To solve the dual integral equations, the jumps of displacements across crack surfaces are directly expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present near crack tips. The non-local elastic solutions yield a finite stress at crack tips, thus allowing us to use the maximum stress as a fracture criterion. Numerical examples are provided to show the effects of the crack length, the parameter describing functionally graded materials, the lattice parameter of materials and the material constants upon the stress fields near crack tips.     

Thermoelectroelastic response of a functionally graded piezoelectric strip with two parallel axisymmetric cracks

A mixed-mode thermoelectroelastic fracture problem of a functionally graded piezoelectric material strip containing two parallel axisymmetric cracks, such as penny-shaped or annular cracks, is considered in this study. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip and that the strip is under thermal loading. The crack faces are supposed to be insulated thermally and electrically. Using integral transform techniques, the problem is reduced to that of solving two systems of singular integral equations. Systematic numerical calculations are carried out, and the variations of the stress and electric displacement intensity factors are plotted for various values of dimensionless parameters representing the crack size, the crack location and the material non-homogeneity.     

Periodically distributed parallel cracks in a functionally graded piezoelectric (FGP) strip bonded to a FGP substrate under static electromechanical load

In this paper, the Fourier integral transform–singular integral equation method is presented for the problem of a periodic array of cracks in a functionally graded piezoelectric strip bonded to a different functionally graded piezoelectric material. The properties of two materials, such as elastic modulus, piezoelectric constant and dielectric constant, are assumed in exponential forms and vary along the crack direction. The crack surface condition is assumed to be electrically impermeable or permeable. The mixed boundary value problem is reduced to a singular integral equation over crack by applying the Fourier transform and the singular integral equation is solved numerically by using the Lobatto–Chebyshev integration technique. The analytic expressions of the stress intensity factors and the electric displacement intensity factors are derived. The effects of the loading parameter λ, material constants and the geometry parameters on the stress intensity factor, the energy release ratio and the energy density factor are studied.     

Near-tip fields and intensity factors for interfacial cracks in dissimilar anisotropic piezoelectric media

A complete form of stress and electric displacement fields in the vicinity of the tip of an interfacial crack, between two dissimilar anisotropic piezoelectric media, is derived by using the complex function theory. New definitions of real-valued stress and electric displacement intensity factors for the interfacial crack are proposed. These definitions are extensions of those for cracks in homogeneous piezoelectric media. Closed form solutions of the stress and electric displacement intensity factors for a semi-infinite crack as well as for a finite crack at the interface between two dissimilar piezoelectric media are also obtained by using the mutual integral.     

Mutual influence of two parallel coaxial cracks in a composite material with initial stresses

By using the approaches of three-dimensional linearized mechanics of deformed bodies, we study the axisymmetric problem of two parallel coaxial circular cracks in an infinite composite material with initial stresses acting in the planes of the cracks. The resolving system of Fredholm integral equations of the second kind is deduced and the representations of the stress intensity factors in the vicinity of the crack tips are obtained. The dependences of the stress intensity factors on the initial stresses and the distance between the cracks are determined. For two types of composite materials, namely, for a layered composite with isotropic layers and a composite with stochastic reinforcement by short ellipsoidal fibers, the stress intensity factors are computed and their dependences on the initial stresses, physicomechanical characteristics of the composites, and geometric parameters of the problem are investigated. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 4, pp. 58–67, July–August, 2008.     

Cylindrical interface cracks in 1-3 piezocomposites

1-3 Piezocomposites are made by embedding piezoelectric fibers/rods in polymer matrix materials. Fiber–matrix interface fracture can affect the performance of piezocomposites. In this paper, axisymmetric interfacial cracks in piezocomposites are studied by considering an idealized model of a single piezoelectric fiber in a matrix material. The displacement discontinuity method is used to formulate the Mode I and II crack problems. The fundamental solutions required for DDM are derived explicitly by using the electroelastic field equations and Fourier integral transforms. The dependence of Mode I and II stress intensity factors of single and multiple interface cracks on fiber and matrix material properties, crack length and distance between cracks are investigated.     

Dynamic behavior of a crack in a functionally graded piezoelectric strip bonded to two dissimilar half piezoelectric material planes

Summary. The dynamic behavior of a crack in a functionally graded piezoelectric material (FGPM) strip bonded to two half dissimilar piezoelectric material planes subjected to combined harmonic anti-plane shear wave and in-plane electrical loading was studied under the limited permeable and permeable electric boundary conditions. It was assumed that the elastic stiffness, piezoelectric constant and dielectric permittivity of the functionally graded piezoelectric layer vary continuously along the thickness of the strip. By using the Fourier transform, the problem can be solved with a set of dual integral equations in which the unknown variables are the jumps of the displacements and the electric potentials across the crack surfaces. In solving the dual integral equations, the jumps of the displacements and the electric potentials across the crack surfaces were expanded in a series of Jacobi polynomials. Numerical results illustrate the effects of the gradient parameter of FGPM, electric loading, wave number, thickness of FGPM strip and electric boundary conditions on the dynamic stress intensity factors (SIFs).     

Determination of stress field in an elastic solid weakened by parallel penny-shaped cracks

Summary This paper investigates the stress intensity factors of two penny-shaped cracks with different sizes in a three-dimensional elastic solid under uniaxial tension. The two cracks are symmetrically parallel located in the isotropic solid. Based on Eshelby's equivalent inclusion method and the superposition pronciple of the elasticity theory, a closed-form analytical elastic solution for the stress intensity factors (SIFs) on the boundaries of the cracks is obtained when the center distance between the two cracks is much larger than the crack sizes. A numerical method is employed to extract the solution for small center distance case. It is found that, due to the interaction between the two cracks, the first and second kinds of SIFs exist at the same time even if the applied stress is pure tension. Numerical examples are given for different configurations and it is clearly shown that the SIFs are strongly determined by the distance between the centers of the two cracks.     

Interaction of parallel cracks across the interface of materials

We reduce a three-dimensional problem of the theory of elasticity about the interaction of cracks in a body to the solution of a system of boundary integral equations for functions that characterize mutual displacements of the opposite surfaces of the cracks in the process of deformation of the body. For the case of parallel cracks, the regular kernels of these equations taking into account the interaction of cracks across the interface of materials are presented in the explicit form. The deduced boundary integral equations are solved by using a numerical-analytic approach. For two disk-shaped parallel cracks in different materials loaded by normal forces, we obtain the dependences of the stress intensity factors on the ratio of elastic constants of the components of the body for various distances from the crack to the interface of materials. State University “L'viv'ska Politekhnika,” L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 35, No. 2, pp. 12–20, March–April, 1999.     

Numerical simulation of bi-material interfacial cracks using EFGM and XFEM

In this paper, bi-material interfacial cracks have been simulated using element free Galerkin method (EFGM) and extended finite element method (XFEM) under mode-I and mixed mode loading conditions. Few crack interaction problems of dissimilar layered materials are also simulated using extrinsic partition of unity enriched approach. Material discontinuity has been modeled by a signed distance function whereas strong discontinuity has been modeled by two functions i.e. Heaviside and asymptotic crack tip enrichment functions. The stress intensity factors for bi-material interface cracks are numerically evaluated using the modified domain form of interaction integral. The results obtained by EFGM and XFEM for bi-material edge and center cracks are compared with those available in literature. In order to check the validity of simulations, the results have been obtained for two different ratio of Young’s modulus.     

Dynamic stress intensity factors around two parallel cracks in an infinite elastic plate

Summary Dynamic stresses around two parallel cracks in an infinite elastic plate are obtained. An incoming shock stress wave impinges on the cracks at right angles to their faces. The Fourier-Laplace transform technique is utilized to reduce the problem to dual integral equations. To solve these equations, the differences in the crack surface displacements are expanded in a series of functions which are zero outside the cracks. The unknown coefficients occurring in those series are solved using the Schmidt method. The stress intensity factors defined in the Laplace transform domain are inverted numerically, in the physical space.     

Modelling and analysis of the dynamic behaviour of piezoelectric materials containing interacting cracks

This study is concerned with the treatment of the dynamic behaviour of piezoelectric materials containing interacting cracks under antiplane mechanical and inplane electric loading. A general electrical boundary condition is used to enable the treatment of both permeable and impermeable conditions along the crack surfaces. The theoretical solution of the problem is formulated using integral transform techniques and an appropriate pseudo-incident wave method. The resulting singular integral equations are solved using Chebyshev polynomials to provide the dynamic stress and electric fields. Numerical examples are provided to show the effect of the geometry of the cracks, the piezoelectric constant of the material and the frequency of the incident wave upon the dynamic stress intensity factors. The results show the significant effect of electromechanical coupling upon local stress distribution.     

Conducting cracks in dissimilar piezoelectric media

Complete stress and electric fields near the tip of a conducting crack between two dissimilar anisotropic piezoelectric media, are obtained in terms of two generalized bimaterial matrices proposed in this paper. It is shown that the general interfacial crack-tip field consists of two pairs of oscillatory singularities. New definitions of real-valued stress and electric field intensity factors are proposed. Exact solutions of the stress and electric fields for basic interface crack problems are obtained. An alternate form of the J integral is derived, and the mutual integral associated with the J integral is proposed. Closed form solutions of the stress and electric field intensity factors due to electromechanical loading and the singularities for a semi-infinite crack as well as for a finite crack at the interface between two dissimilar piezoelectric media, are also obtained by using the mutual integral.     

Dynamic response for non-homogeneous piezoelectric medium with multiple cracks

A general procedure to analyze the dynamic response of non-homogeneous piezoelectric medium containing some non-collinear cracks is developed. It is assumed that all the material properties only depend on the coordinates y (along the thickness direction). The assumption is made that the non-homogeneous medium is composed of numerous laminae with their surfaces perpendicular to the thick direction. The solution method is based upon the Fourier and Laplace transforms to reduce the boundary value problem to a system of generalized singularity equations in the Laplace transform domain. The singular integral equations for the problem are derived and numerically solved by weight residual value method. The time-dependent full field solutions are obtained in the time domain. As numerical illustration, the stress and electric displacement intensity factors for a three-layer plate specimen with two cracks are presented. It is found that the stress and electric fields are coupled in the crack plane ahead of the crack tip for non-homogenous piezoelectric materials.     

The transient response of bonded piezoelectric and elastic half space with multiple interfacial collinear cracks

Summary This paper investigates the dynamic behavior of a bonded piezoelectric and elastic half space containing multiple interfacial collinear cracks subjected to transient electro-mechanical loads. Both the permeable and impermeable boundary conditions are examined and discussed. Based on the use of integral transform techniques, the problem is reduced to a set of singular integral equations, which can be solved using Chebyshev polynomial expansions. Numerical results are provided to show the effect of the geometry of interacting collinear cracks, the applied electric fields, and the electric boundary conditions along the crack faces on the resulting dynamic stress intensity and electric displacement intensity factors.     

Scattering of the harmonic anti-plane shear waves by a crack in functionally graded piezoelectric materials

In the present paper, the dynamic behavior of a Griffth crack in the functionally graded piezoelectric material (FGPM) is investigated. It is assumed that the elastic stiffness, piezoelectric constant, dielectric permittivity and mass density of the FGPM vary continuously as an exponential function, and that FGPM is under the anti-plane mechanical loading and in-plane electrical loading. By using the Fourier transform and defining the jumps of displacement and electric potential components across the crack surface as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement and electric potential components across the crack surface are expanded in a series of Jacobi polynomial. Numerical examples are provided to show the effects of material properties on the stress and the electric displacement intensity factors.     

Dynamic antiplane behaviour of interacting cracks in a piezoelectric medium

In this article, we examine the dynamic interaction between two cracks in a piezoelectric medium under incident antiplane shear wave loading. The theoretical formulations governing the steady-state problem are based upon the use of integral transform techniques and a self-consistent iterative method. The resulting dynamic stress intensity factors at the interacting cracks are obtained by solving the appropriate singular integral equations using Chebyshev Polynomials at different loading frequencies. Numerical examples are provided to show the effect of the geometry of the cracks, the piezoelectric constants of the material and the frequency of the incident wave upon the dynamic stress intensity factor of the cracks. This revised version was published online in August 2006 with corrections to the Cover Date.