Electroelastic Wave Scattering in a Cracked Dielectric Polymer under a Uniform Electric Field

  • Yasuhide Shindo1Email author and

    Affiliated with

    • Fumio Narita1

      Affiliated with

      Boundary Value Problems20092009:949124

      DOI: 10.1155/2009/949124

      Received: 25 April 2009

      Accepted: 18 May 2009

      Published: 16 June 2009

      Abstract

      We investigate the scattering of plane harmonic compression and shear waves by a Griffith crack in an infinite isotropic dielectric polymer. The dielectric polymer is permeated by a uniform electric field normal to the crack face, and the incoming wave is applied in an arbitrary direction. By the use of Fourier transforms, we reduce the problem to that of solving two simultaneous dual integral equations. The solution of the dual integral equations is then expressed in terms of a pair of coupled Fredholm integral equations of the second kind having the kernel that is a finite integral. The dynamic stress intensity factor and energy release rate for mode I and mode II are computed for different wave frequencies and angles of incidence, and the influence of the electric field on the normalized values is displayed graphically.

      1. Introduction

      Elastic dielectrics such as insulating materials have been reported to have poor mechanical properties. Mechanical failure of insulators is also a well-known phenomenon. Therefore, understanding the fracture behavior of the elastic dielectrics will provide useful information to the insulation designers. Toupin [1] considered the isotropic elastic dielectric material and obtained the form of the constitutive relations for the stress and electric fields. Kurlandzka [2] investigated a crack problem of an elastic dielectric material subjected to an electrostatic field. Pak and Herrmann [3, 4] also derived a material force in the form of a path-independent integral for the elastic dielectric medium, which is related to the energy release rate. Recently, Shindo and Narita [5] considered the planar problem for an infinite dielectric polymer containing a crack under a uniform electric field, and discussed the stress intensity factor and energy release rate under mode I and mode II loadings.

      This paper investigates the scattering of in-plane compressional (P) and shear (SV) waves by a Griffith crack in an infinite dielectric polymer permeated by a uniform electric field. The electric field is normal to the crack surface. Fourier transforms are used to reduce the problem to the solution of two simultaneous dual integral equations. The solution of the integral equations is then expressed in terms of a pair of coupled Fredholm integral equations of the second kind. In literature, there are two derivations of dual integral equations. One is the one mentioned in this paper. The other one is for the dual boundary element methods (BEM) [6, 7]. Numerical calculations are carried out for the dynamic stress intensity factor and energy release rate under mode I and mode II, and the results are shown graphically to demonstrate the effect of the electric field.

      2. Basic Equations

      Consider the rectangular Cartesian coordinate system with axes http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq2_HTML.gif . We decompose the electric field intensity vector http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq3_HTML.gif , the polarization vector http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq4_HTML.gif , and the electric displacement vector http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq5_HTML.gif into those representing the rigid body state, indicated by overbars, and those for the deformed state, denoted by lower case letters:

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ1_HTML.gif
      (2.1)

      We assume that the deformation will be small even with large electric fields, and the second terms will have only a minor influence on the total fields. The formulations will then be linearized with respect to these unknown deformed state quantities.

      The linearized field equations are obtained as

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ2_HTML.gif
      (2.2)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq6_HTML.gif is the displacement vector, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq7_HTML.gif is the local stress tensor, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq8_HTML.gif is the mass density, a comma followed by an index denotes partial differentiation with respect to the space coordinate http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq9_HTML.gif or the time http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq10_HTML.gif , and the summation convention for repeated indices is applied.

      The linearized constitutive equations can be written as

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ3_HTML.gif
      (2.3)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq11_HTML.gif is the Maxwell stress tensor, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq12_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq13_HTML.gif are the Lamé constants, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq14_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq15_HTML.gif are the electrostrictive coefficients, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq16_HTML.gif is the permittivity of free space, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq17_HTML.gif = 1 + http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq18_HTML.gif is the specific permittivity, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq19_HTML.gif is the electric susceptibility, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq20_HTML.gif is the Kronecker delta.

      The linearized boundary conditions are found as

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ4_HTML.gif
      (2.4)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq21_HTML.gif is an outer unit vector normal to an undeformed body, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq22_HTML.gif is the permutation symbol, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq23_HTML.gif means the jump in any field quantity http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq24_HTML.gif across the discontinuity surface.

      3. Problem Statement

      Let a Griffith crack be located in the interior of an infinite elastic dielectric. We consider a rectangular Cartesian coordinate system http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq25_HTML.gif such that the crack is placed on the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq26_HTML.gif -axis from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq27_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq28_HTML.gif as shown in Figure 1, and assume that plane strain is perpendicular to the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq29_HTML.gif -axis. A uniform electric field http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq30_HTML.gif is applied perpendicular to the crack surface. For convenience, all electric quantities outside the solid will be denoted by the superscript http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq31_HTML.gif . The solution for the rigid body state is
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Fig1_HTML.jpg
      Figure 1

      Scattering of waves in a dielectric medium with a Griffith crack.

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ5_HTML.gif
      (3.1)

      The equations of motion are given by

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ6_HTML.gif
      (3.2)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq32_HTML.gif is the two-dimensional Laplace operator in the variables http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq33_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq34_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq35_HTML.gif is the Poisson's ratio, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq36_HTML.gif is the shear wave velocity, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq37_HTML.gif . The electric field equations for the perturbed state are

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ7_HTML.gif
      (3.3)

      The electric field equations (3.3) are satisfied by introducing an electric potential http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq38_HTML.gif such that

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ8_HTML.gif
      (3.4)

      The displacement components can be written in terms of two scalar potentials http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq39_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq40_HTML.gif as

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ9_HTML.gif
      (3.5)

      The equations of motion become

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ10_HTML.gif
      (3.6)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq41_HTML.gif is the compression wave velocity.

      Let an incident plane harmonic compression wave (P-wave) be directed at an angle http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq42_HTML.gif with the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq43_HTML.gif -axis so that

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ11_HTML.gif
      (3.7)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq44_HTML.gif is the amplitude of the incident P-wave, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq45_HTML.gif is the circular frequency. The superscript http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq46_HTML.gif stands for the incident component. Similarly, if an incident plane harmonic shear wave (SV-wave) impinges on the crack at an angle http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq47_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq48_HTML.gif -axis, then

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ12_HTML.gif
      (3.8)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq49_HTML.gif is the amplitude of the incident SV-wave. In view of the harmonic time variation of the incident waves given by (3.7) and (3.8), the field quantities will all contain the time factor exp http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq50_HTML.gif which will henceforth be dropped.

      The problem may be split into two parts: one symmetric (opening mode, Mode I) and the other skew-symmetric (sliding mode, Mode II). Hence, the boundary conditions for the scattered fields are

      Mode I:

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ13_HTML.gif
      (3.9)

      Mode II:

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ14_HTML.gif
      (3.10)

      where the subscript http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq51_HTML.gif and 2 correspond to the incident P- and SV-waves, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq52_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq53_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq54_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq55_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq56_HTML.gif and, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq57_HTML.gif are the compression and shear wave numbers, respectively, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq58_HTML.gif .

      4. Method of Solution

      The desired solution of the original problem can be obtained by superposition of the solutions for the two cases: mode I and mode II. The problem will further be divided into two parts: (1) symmetric with respect to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq59_HTML.gif and (2) antisymmetric with respect to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq60_HTML.gif .

      4.1. Mode I Problem

      4.1.1. Symmetric Solution for Mode I Crack

      The boundary conditions for symmetric scattered fields can be written as
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ15_HTML.gif
      (4.1)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ16_HTML.gif
      (4.2)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ17_HTML.gif
      (4.3)

      where the subscript http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq61_HTML.gif stands for the symmetric part. It can be shown that solutions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq62_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq63_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq64_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq65_HTML.gif of (3.4) and (3.6) for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq66_HTML.gif are

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ18_HTML.gif
      (4.4)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ19_HTML.gif
      (4.5)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ20_HTML.gif
      (4.6)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq67_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq68_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq69_HTML.gif are unknown functions, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq70_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq71_HTML.gif are

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ21_HTML.gif
      (4.7)

      The functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq72_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq73_HTML.gif should be restricted as

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ22_HTML.gif
      (4.8)

      in the upper half-space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq74_HTML.gif , because of a radiation condition at infinity and an edge condition near the crack tip. A simple calculation leads to the displacement and stress expressions:

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ23_HTML.gif
      (4.9)
      The boundary condition of (4.1) leads to the following relation between unknown functions:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ24_HTML.gif
      (4.10)

      The satisfaction of the two mixed boundary conditions (4.2) and (4.3) leads to two simultaneous dual integral equations of the following form:

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ25_HTML.gif
      (4.11)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ26_HTML.gif
      (4.12)

      in which http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq75_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq76_HTML.gif are known functions given by

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ27_HTML.gif
      (4.13)

      and the original unknowns http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq77_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq78_HTML.gif are related to the new one http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq79_HTML.gif through

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ28_HTML.gif
      (4.14)

      The set of two simultaneous dual integral equations (4.11) and (4.12) may be solved by using a new function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq80_HTML.gif , and the result is

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ29_HTML.gif
      (4.15)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq81_HTML.gif is the zero-order Bessel function of the first kind, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq82_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq83_HTML.gif are

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ30_HTML.gif
      (4.16)

      The function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq84_HTML.gif is governed by the following Fredholm integral equation of second kind:

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ31_HTML.gif
      (4.17)

      where the kernel http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq85_HTML.gif is given by

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ32_HTML.gif
      (4.18)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ33_HTML.gif
      (4.19)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ34_HTML.gif
      (4.20)

      The kernel function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq86_HTML.gif (4.18) is an infinite integral that has a rather slow of convergence. To improve this problem the infinite integral is converted into integrals with finite limits. Thus, for the calculation of the integral, we consider the contour integrals

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ35_HTML.gif
      (4.21)
      where the contours http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq87_HTML.gif are defined in Figure 2, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq88_HTML.gif are, respectively, the zero-order Hankel functions of the first and second kinds, and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Fig2_HTML.jpg
      Figure 2

      The counters of integration.

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ36_HTML.gif
      (4.22)

      The integrands in (4.21) satisfy Jordan's lemma on the infinite quarter circles, so that,

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ37_HTML.gif
      (4.23)

      where

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ38_HTML.gif
      (4.24)

      Because of the second of (4.8), the integral in (4.18) must be taken along a path located slightly below the real http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq89_HTML.gif -axis as in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq90_HTML.gif . Therefore http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq91_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq92_HTML.gif can be finally written as

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ39_HTML.gif
      (4.25)

      where

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ40_HTML.gif
      (4.26)

      The kernel http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq93_HTML.gif is symmetric in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq94_HTML.gif , and the value of this kernel for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq95_HTML.gif is obtained by interchanging http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq96_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq97_HTML.gif in (4.25).

      4.1.2. Antisymmetric Solution for Mode I Crack

      The boundary conditions for anti-symmetric scattered fields can be written as
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ41_HTML.gif
      (4.27)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ42_HTML.gif
      (4.28)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ43_HTML.gif
      (4.29)

      where the subscript http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq98_HTML.gif stands for the anti-symmetric part. The solutions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq99_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq100_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq101_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq102_HTML.gif are

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ44_HTML.gif
      (4.30)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ45_HTML.gif
      (4.31)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ46_HTML.gif
      (4.32)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq103_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq104_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq105_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq106_HTML.gif are unknown functions. The displacements and stresses are obtained as

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ47_HTML.gif
      (4.33)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ48_HTML.gif
      (4.34)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ49_HTML.gif
      (4.35)

      The relation between unknown functions can be found by the same procedure as in the symmetric case. The boundary condition of (4.27) leads to the following relation:

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ50_HTML.gif
      (4.36)

      The boundary conditions in (4.28) and (4.29) lead to two simultaneous dual integral equations of the following form:

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ51_HTML.gif
      (4.37)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ52_HTML.gif
      (4.38)

      in which the original unknowns http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq107_HTML.gif are related to the new one http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq108_HTML.gif through

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ53_HTML.gif
      (4.39)

      The unknowns http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq109_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq110_HTML.gif can be found by the same method of approach as in the symmetric case. The results are

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ54_HTML.gif
      (4.40)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq111_HTML.gif is the first-order Bessel function of the first kind, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq112_HTML.gif in (4.40) is the solution of the following Fredholm integral equation of the second kind:

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ55_HTML.gif
      (4.41)

      where

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ56_HTML.gif
      (4.42)

      By using the contours of integration in Figure 2, the kernel http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq113_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq114_HTML.gif can be rewritten in the form

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ57_HTML.gif
      (4.43)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq115_HTML.gif is the first-order Hankel function of the first kind. The value of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq116_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq117_HTML.gif is obtained by interchanging http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq118_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq119_HTML.gif in (4.43).

      4.1.3. Mode I Dynamic Singular Stresses Near the Crack Tip

      The mode I dynamic electric stress intensity factor http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq120_HTML.gif is

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ58_HTML.gif
      (4.44)

      where

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ59_HTML.gif
      (4.45)

      Next, we examine the static electroelastric crack problem. The boundary conditions may be written as

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ60_HTML.gif
      (4.46)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ61_HTML.gif
      (4.47)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ62_HTML.gif
      (4.48)

      The electric stress intensity factor http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq121_HTML.gif may be obtained as

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ63_HTML.gif
      (4.49)

      The dynamic stress intensity factor http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq122_HTML.gif can be found as

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ64_HTML.gif
      (4.50)

      The dynamic electroelastic stress is given by

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ65_HTML.gif
      (4.51)

      The singular parts of the dynamic local stresses and Mexwell stresses near the crack tip can be expressed as

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ66_HTML.gif
      (4.52)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ67_HTML.gif
      (4.53)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq123_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq124_HTML.gif are the polar coordinates. Also, the singular parts of the displacements and electric fields near the crack tip are

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ68_HTML.gif
      (4.54)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ69_HTML.gif
      (4.55)

      4.2. Mode II Problem

      Since the mode II problem may also be reduced to the solution of two simultaneous dual integral equations in the same way as the mode I, many of the details of solution procedure will be omitted and only the essential steps will be provided.

      4.2.1. Symmetric Solution for Mode II Crack

      The boundary conditions for symmetric scattered fields are
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ70_HTML.gif
      (4.56)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ71_HTML.gif
      (4.57)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ72_HTML.gif
      (4.58)

      Replace the subscript http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq125_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq126_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq127_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq128_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq129_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq130_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq131_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq132_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq133_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq134_HTML.gif , respectively, in (4.30)–(4.35). The boundary condition of (4.56) leads to

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ73_HTML.gif
      (4.59)

      Introducing the abbreviation

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ74_HTML.gif
      (4.60)

      and in view of two mixed boundary conditions (4.57) and (4.58), together with (4.59) and (4.60), we have the following two simultaneous dual integral equations for the determination of the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq135_HTML.gif :

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ75_HTML.gif
      (4.61)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ76_HTML.gif
      (4.62)

      where

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ77_HTML.gif
      (4.63)

      The solution of (4.61) and (4.62) are obtained by using two new functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq136_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq137_HTML.gif , and the results are

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ78_HTML.gif
      (4.64)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq138_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq139_HTML.gif are the solutions of the following Fredholm integral equations of the second kind:

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ79_HTML.gif
      (4.65)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ80_HTML.gif
      (4.66)

      The kernels are given by

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ81_HTML.gif
      (4.67)

      where

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ82_HTML.gif
      (4.68)

      and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq140_HTML.gif . The kernels http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq141_HTML.gif are symmetric in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq142_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq143_HTML.gif .

      4.2.2. Antisymmetric Solution for Mode II Crack

      The boundary conditions for anti-symmetric scattered fields are
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ83_HTML.gif
      (4.69)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ84_HTML.gif
      (4.70)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ85_HTML.gif
      (4.71)

      Let replace the subscript http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq144_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq145_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq146_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq147_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq148_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq149_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq150_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq151_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq152_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq153_HTML.gif in (4.4)–(4.6). The boundary condition of (4.69) leads to

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ86_HTML.gif
      (4.72)

      Introducing the abbreviation

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ87_HTML.gif
      (4.73)

      and in view of boundary conditions (4.70) and (4.71), together with (4.72) and (4.73), we have the following two simultaneous dual integral equations:

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ88_HTML.gif
      (4.74)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ89_HTML.gif
      (4.75)

      Equations (4.74) and (4.75) yield the solutions

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ90_HTML.gif
      (4.76)

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq154_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq155_HTML.gif are the solutions of the following Fredholm integral equations of the second kind:

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ91_HTML.gif
      (4.77)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ92_HTML.gif
      (4.78)

      where

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ93_HTML.gif
      (4.79)

      and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq156_HTML.gif are symmetric in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq157_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq158_HTML.gif .

      4.2.3. Mode II Dynamic Singular Stresses Near the Crack Tip

      The dynamic stress intensity factor http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq159_HTML.gif is obtained as

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ94_HTML.gif
      (4.80)

      where

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ95_HTML.gif
      (4.81)
      The singular parts of the dynamic local stresses and Maxwell stresses near the crack tip can be derived as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ96_HTML.gif
      (4.82)

      The singular parts of the displacements and electric fields near the crack tip can be expressed as

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ97_HTML.gif
      (4.83)

      where

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ98_HTML.gif
      (4.84)

      5. Dynamic Energy Release Rate

      The dynamic energy release rate http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq160_HTML.gif is obtained as

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ99_HTML.gif
      (5.1)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq161_HTML.gif is the region with the contour http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq162_HTML.gif . This expression may be thought of as an extension to the J-integral given in [3]. If all the electrical field quantities are made to vanish, then (5.1) reduces to the dynamic energy release rate for the elastic materials [8]. Writing the dynamic energy release rate expression in terms of the mode I dynamic stress intensity factor, there results

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ100_HTML.gif
      (5.2)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ101_HTML.gif
      (5.3)

      6. Results and Discussion

      To examine the effect of electroelastic interactions on the dynamic stress intensity factor and dynamic energy release rate, the solutions of the Fredholm integral equations of the second kind (4.17), (4.41) for Mode I and (4.65), (4.66), (4.77), (4.78) for Mode II have been computed numerically by the use of Gaussian quadrature formulas. We can consider polymethylmethacrylate (PMMA), and the engineering material constants of PMMA are listed in Table 1. The dynamic stress intensity factor http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq163_HTML.gif can be found as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq164_HTML.gif .
      Table 1

      Material properties of PMMA.

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq165_HTML.gif

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq166_HTML.gif

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq167_HTML.gif

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq168_HTML.gif

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq169_HTML.gif

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq170_HTML.gif

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq171_HTML.gif

      0.4

      0

      3.61

      2

      3

      Figure 3 exhibits the variation of the normalized mode I dynamic stress intensity factor http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq172_HTML.gif against the normalized frequency http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq173_HTML.gif subjected to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq174_HTML.gif -waves for the normalized electric field http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq175_HTML.gif and the angle of incidence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq176_HTML.gif . The dynamic stress intensity factor drops rapidly beyond the first maximum and exhibits oscillations of approximately constant period as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq177_HTML.gif increases. The peak value of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq178_HTML.gif under http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq179_HTML.gif is 1.364. Also, the peak values of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq180_HTML.gif under http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq181_HTML.gif are 1.522, 2.416, 3.310 for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq182_HTML.gif , respectively. As http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq183_HTML.gif ,  the dynamic stress intensity factor tends to static stress intensity factor [5]. In the absence of the electric fields, the dynamic stress intensity factor becomes the solution for the elastic solid (see e.g. [9]). Figure 4 also shows the variation of the normalized mode I dynamic energy release rate http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq184_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq185_HTML.gif is the static energy release rate. The peak values of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq186_HTML.gif under http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq187_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq188_HTML.gif are 1.861, 2.361, 5.838, 10.96, respectively. Figure 5 shows the normalized mode I dynamic stress intensity factor http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq189_HTML.gif versus http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq190_HTML.gif subjected to P-waves for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq191_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq192_HTML.gif . The peak values of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq193_HTML.gif under http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq194_HTML.gif are 1.078, 1.198 for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq195_HTML.gif , respectively. Figure 6 shows the normalized mode II dynamic stress intensity factor http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq196_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq197_HTML.gif versus http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq198_HTML.gif subjected to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq199_HTML.gif -waves for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq200_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq201_HTML.gif . The effect of electric fields on the mode II dynamic stress intensity factor is small. Figure 7 displays the normalized mode I dynamic stress intensity factor http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq202_HTML.gif against the angle of incidence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq203_HTML.gif subjected to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq204_HTML.gif -waves for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq205_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq206_HTML.gif . The mode I dynamic stress intensity factors for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq207_HTML.gif and 0.8 attain its maximum values at an incident angle of approximately http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq208_HTML.gif .
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Fig3_HTML.jpg
      Figure 3

      Mode I dynamic stress intensity factor versus frequency (P-waves, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq209_HTML.gif ).

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Fig4_HTML.jpg
      Figure 4

      Mode I dynamic energy relrase rate versus frequency (P-waves, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq210_HTML.gif ).

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Fig5_HTML.jpg
      Figure 5

      Mode I dynamic stress intensity factor versus frequency (P-waves, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq211_HTML.gif ).

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Fig6_HTML.jpg
      Figure 6

      Mode II dynamic stress intensity factor versus frequency (P-waves, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq212_HTML.gif ).

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Fig7_HTML.jpg
      Figure 7

      Mode I dynamic stress intensity factor versus angle of incidence (P-waves).

      Figure 8 shows the variation of the normalized mode II dynamic stress intensity factor http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq213_HTML.gif versus http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq214_HTML.gif subjected to SV-waves for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq215_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq216_HTML.gif . The electric fields have small effect on the mode II dynamic stress intensity factor. Figure 9 shows the normalized mode I dynamic stress intensity factor http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq217_HTML.gif against http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq218_HTML.gif subjected to SV-waves for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq219_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq220_HTML.gif . Similar trend to the case under P-waves is observed.
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Fig8_HTML.jpg
      Figure 8

      Mode II dynamic stress intensity factor versus frequency (SV-waves, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq221_HTML.gif ).

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Fig9_HTML.jpg
      Figure 9

      Mode I dynamic stress intensity factor versus frequency (SV-waves, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq222_HTML.gif ).

      7. Conclusions

      The dynamic electroelastic problem for a dielectric polymer having a finite crack has been analyzed theoretically. The results are expressed in terms of the dynamic stress intensity factor and dynamic energy release rate. It is found that the dynamic stress intensity factor and dynamic energy release rate tend to increase with frequency reaching a peak and then decrease in magnitude. These peaks depend on the angle of incidence. Also, applied electric fields increase the mode I dynamic stress intensity factor and dynamic energy release rate, whereas the mode II dynamic stress intensity factor is less dependent on the electric field.

      Authors’ Affiliations

      (1)
      Department of Materials Processing, Graduate School of Engineering, Tohoku University

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      Copyright

      © Y. Shindo and F. Narita. 2009

      This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.