Open Access

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

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq1_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq3_HTML.gif , the polarization vector https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq4_HTML.gif , and the electric displacement vector https://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:

https://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

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq6_HTML.gif is the displacement vector, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq7_HTML.gif is the local stress tensor, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq9_HTML.gif or the time https://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

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

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

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

where https://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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq22_HTML.gif is the permutation symbol, and https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq26_HTML.gif -axis from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq27_HTML.gif to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq29_HTML.gif -axis. A uniform electric field https://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 https://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
https://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.

https://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

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

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq33_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq34_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq35_HTML.gif is the Poisson's ratio, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq36_HTML.gif is the shear wave velocity, and https://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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq38_HTML.gif such that

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq39_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq40_HTML.gif as

https://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

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

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq42_HTML.gif with the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq43_HTML.gif -axis so that

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

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq45_HTML.gif is the circular frequency. The superscript https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq47_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq48_HTML.gif -axis, then

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

where https://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 https://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:

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

Mode II:

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

where the subscript https://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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq52_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq53_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq54_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq55_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq56_HTML.gif and, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq59_HTML.gif and (2) antisymmetric with respect to https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ15_HTML.gif
(4.1)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ16_HTML.gif
(4.2)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ17_HTML.gif
(4.3)

where the subscript https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq62_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq63_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq64_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq66_HTML.gif are

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

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

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

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

https://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 https://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:

https://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:
https://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:

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

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

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

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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq80_HTML.gif , and the result is

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

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq82_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq83_HTML.gif are

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

The function https://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:

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

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

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

The kernel function https://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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ35_HTML.gif
(4.21)
where the contours https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq87_HTML.gif are defined in Figure 2, https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Fig2_HTML.jpg
Figure 2

The counters of integration.

https://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,

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

where

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq89_HTML.gif -axis as in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq90_HTML.gif . Therefore https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq91_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq92_HTML.gif can be finally written as

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

where

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

The kernel https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq93_HTML.gif is symmetric in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq95_HTML.gif is obtained by interchanging https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq96_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ41_HTML.gif
(4.27)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ42_HTML.gif
(4.28)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ43_HTML.gif
(4.29)

where the subscript https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq99_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq100_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq101_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq102_HTML.gif are

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq103_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq104_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq105_HTML.gif , and https://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

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

https://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:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ51_HTML.gif
(4.37)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq107_HTML.gif are related to the new one https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq108_HTML.gif through

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

The unknowns https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq109_HTML.gif and https://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

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

where https://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 https://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:

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

where

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq113_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq114_HTML.gif can be rewritten in the form

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

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq116_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq117_HTML.gif is obtained by interchanging https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq118_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq120_HTML.gif is

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

where

https://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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ60_HTML.gif
(4.46)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ61_HTML.gif
(4.47)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq121_HTML.gif may be obtained as

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq122_HTML.gif can be found as

https://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

https://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

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq123_HTML.gif and https://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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ68_HTML.gif
(4.54)
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ70_HTML.gif
(4.56)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ71_HTML.gif
(4.57)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ72_HTML.gif
(4.58)

Replace the subscript https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq125_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq126_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq127_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq128_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq129_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq130_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq131_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq132_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq133_HTML.gif , and https://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

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

Introducing the abbreviation

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq135_HTML.gif :

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

where

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq136_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq137_HTML.gif , and the results are

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq138_HTML.gif and https://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:

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

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

where

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

and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq140_HTML.gif . The kernels https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq141_HTML.gif are symmetric in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq142_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ83_HTML.gif
(4.69)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ84_HTML.gif
(4.70)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ85_HTML.gif
(4.71)

Let replace the subscript https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq144_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq145_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq146_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq147_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq148_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq149_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq150_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq151_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq152_HTML.gif and https://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

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

Introducing the abbreviation

https://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:

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

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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq154_HTML.gif and https://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:

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

where

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

and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq156_HTML.gif are symmetric in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq157_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq159_HTML.gif is obtained as

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

where

https://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:
https://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

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

where

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq160_HTML.gif is obtained as

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq161_HTML.gif is the region with the contour https://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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_Equ100_HTML.gif
(5.2)
where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq163_HTML.gif can be found as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq164_HTML.gif .
Table 1

Material properties of PMMA.

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

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

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

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

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

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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq172_HTML.gif against the normalized frequency https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq173_HTML.gif subjected to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq174_HTML.gif -waves for the normalized electric field https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq175_HTML.gif and the angle of incidence https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq177_HTML.gif increases. The peak value of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq178_HTML.gif under https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq180_HTML.gif under https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq182_HTML.gif , respectively. As https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq184_HTML.gif , where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq186_HTML.gif under https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq187_HTML.gif for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq189_HTML.gif versus https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq190_HTML.gif subjected to P-waves for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq191_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq192_HTML.gif . The peak values of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq193_HTML.gif under https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq194_HTML.gif are 1.078, 1.198 for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq196_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq197_HTML.gif versus https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq198_HTML.gif subjected to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq199_HTML.gif -waves for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq200_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq202_HTML.gif against the angle of incidence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq203_HTML.gif subjected to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq204_HTML.gif -waves for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq205_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq208_HTML.gif .
https://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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq209_HTML.gif ).

https://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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq210_HTML.gif ).

https://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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq211_HTML.gif ).

https://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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq212_HTML.gif ).

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq213_HTML.gif versus https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq214_HTML.gif subjected to SV-waves for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq215_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq217_HTML.gif against https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq218_HTML.gif subjected to SV-waves for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq219_HTML.gif and https://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.
https://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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F949124/MediaObjects/13661_2009_Article_890_IEq221_HTML.gif ).

https://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, https://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.