Scattering of harmonic anti-plane shear waves by two collinear cracks in functionally graded piezoelectric materials

doi:10.1016/j.euromechsol.2004.03.002

European Journal of Mechanics - A/Solids

Volume 23, Issue 4, July–August 2004, Pages 633-643

https://doi.org/10.1016/j.euromechsol.2004.03.002 Get rights and content

Abstract

In the present paper, the dynamic behavior of two collinear cracks in the functionally graded piezoelectric material (FGPM) is investigated. It is assumed that the elastic moduli, 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 electric loading. By using Fourier transform and defining the jumps of displacement and electric potential 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 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 electric displacement intensity factors.

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Cited by (35)

Interaction of lateral cracks in double scratching of single-crystal silicon carbide
2019, Theoretical and Applied Fracture Mechanics
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The interaction problem can be simplified as one single crack problem under the given tractions and the tractions induced by the other crack. Besides, dual integral equations are generally produced when calculating the SIFs at the crack tip [6,16–19]. The SIFs of two collinear cracks in an infinite elastic solid subjected to a time-harmonic disturbance and cracks in a half plane subjected to impact loads were obtained, respectively, using dual integral equations by Itous [6,16].
In this paper, a new method is established to analyze the interaction of lateral cracks in double scratching process, which is a basic problem in machining of single-crystal silicon carbide. Applying superposition principle, this interaction problem is reduced to the calculation of stress intensity factors (SIFs) around two scratching-induced collinear cracks under scratching-induced stress field. First, values of SIFs at crack tips under the scratching-induced stress field are obtained applying Fourier transformation, gap functions of displacements and Jacobi polynomials. Then, the scratching-induced stress filed is obtained by the superposition of Boussinesq field, Cerruti field and Blister stress field. Finally, this calculation method is verified by comparing with the results in literature in special case. It is found that values of SIFs decrease dramatically when scratching separation distance increases. The interaction effect can be neglected when the separation distance is larger than 1.6 times of the length of lateral cracks generated in single scratching.
Mixed mode transient analysis of functionally graded piezoelectric plane weakened by multiple cracks
2019, Theoretical and Applied Fracture Mechanics
In this paper, a functionally graded piezoelectric (FGP) plane weakened by multiple parallel cracks is investigated under mixed mode mechanical and electrical impacts. The system contains Volterra type glide and climb edge dislocations. The material properties of the system are assumed to vary exponentially along the x-axis. At first, the stress field and electric displacement in FGP plane are derived by using a single dislocation. Then, a system of singular integral equations with a simple Cauchy kernel of the transient dynamic problem is obtained by determining a distributed dislocation density on the crack surface and using the Fourier and Laplace integral transforms, which are solved numerically using the Lobatto-Chebyshev integration formula. The dynamic field intensity factors are determined through numerical Laplace inversion and the distributed dislocation technique (DDT). Finally, various examples are provided to investigate the influence of the material properties, electromechanical coupling factor, loading conditions and cracks arrangement on the dynamic fracture behavior of the interacting cracks.
Several embedded cracks in a functionally graded piezoelectric strip under dynamic loading
2018, Computers and Mathematics with Applications
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There has been a plethora of research regarding these materials in the field of fracture mechanics. It, also, should be noted that because of their diverse applications, piezoelectric materials have been studied under various types of loadings namely; static [3], time-harmonic [4–6] and transient [7] loadings types. Fracture analysis of piezoelectric materials usually contains two kinds of boundary conditions known as the electrically impermeable and permeable on crack faces [8].
This paper deals with the fracture analysis of multiple cracks in a functionally graded piezoelectric (FGP) layer under applied anti-plane time-harmonic mechanical and in-plane electrical loading within the framework of linear electro-elasticity. The analysis is based on the stress and electric fields caused by Volterra-type screw dislocation in the medium. The crack surfaces are assumed to be impermeable condition. The material properties of the medium are considered to obey exponential variations. In order to model the cracked layer, the distributed dislocation technique is employed to perform Cauchy-type singular integral equations for layer, in which the unknown variables are dislocation densities. The dislocation densities are then employed to derive field intensity factors at the crack tips. The results show that the stress and the electric displacement intensity factors at the crack tips depend on the cracks configuration, frequency and material properties. Finally, several numerical examples are solved to obtain the dynamic stress intensity factors (DSIFs) and electric displacement intensity factors.
Dynamic behavior of several cracks in functionally graded strip subjected to anti-plane time-harmonic concentrated loads
2013, Acta Mechanica Solida Sinica
This paper contains a theoretical formulations and solutions of multiple cracks subjected to an anti-plane time-harmonic point load in a functionally graded strip. The distributed dislocation technique is used to construct integral equations for a functionally graded material strip weakened by several cracks under anti-plane time-harmonic load. These equations are of Cauchy singular type at the location of dislocation, which are solved numerically to obtain the dislocation density on the faces of the cracks. The dislocation densities are employed to evaluate the stress intensity factor and strain energy density factors (SEDFs) for multiple cracks with different configurations. Numerical calculations are presented to show the effects of material properties and the crack configuration on the dynamic stress intensity factors and SEDFs of the functionally graded strip with multiple curved cracks.
Analysis of cracked functionally graded piezoelectric strip
2013, International Journal of Solids and Structures
Citation Excerpt :
Therefore, these materials have been the subject of many researches in the field of fracture mechanics. Different loadings can be recognized in various applications of piezoelectric materials including static (Chen et al., 2003), time-harmonic (Ma et al., 2004, 2005a; Narita and Shindo, 1999) and transient (Chen et al., 2004) types. The piezoelectric materials may be homogeneous or nonhomogeneous.
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.
Elastodynamic analysis of a functionally graded half-plane with multiple sub-surface cracks
2012, Acta Mechanica Solida Sinica
The stress fields are obtained for a functionally graded half-plane containing a Volterra screw dislocation. The elastic shear modulus of the medium is considered to vary exponentially. The dislocation solution is utilized to formulate integral equations for the half-plane weakened by multiple smooth cracks under anti-plane deformation. The integral equations are of Cauchy singular type at the location of dislocation which are solved numerically. Several examples are solved and the stress intensity factors are obtained.

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Scattering of harmonic anti-plane shear waves by two collinear cracks in functionally graded piezoelectric materials

Abstract

Theoret. Appl. Fract. Mech.

Mech. Res. Commun.

Theoret. Appl. Fract. Mech.

Mech. Res. Commun.

Theoret. Appl. Fract. Mech.

Int. J. Engrg. Sci.

Materials and Design

Table of Integral, Series and Products