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

## 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 and . We decompose the electric field intensity vector , the polarization vector , and the electric displacement vector into those representing the rigid body state, indicated by overbars, and those for the deformed state, denoted by lower case letters:

(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

(2.2)

where is the displacement vector, is the local stress tensor, is the mass density, a comma followed by an index denotes partial differentiation with respect to the space coordinate or the time , and the summation convention for repeated indices is applied.

The linearized constitutive equations can be written as

(2.3)

where is the Maxwell stress tensor, and are the LamÃ© constants, and are the electrostrictive coefficients, is the permittivity of free space, = 1 + is the specific permittivity, is the electric susceptibility, and is the Kronecker delta.

The linearized boundary conditions are found as

(2.4)

where is an outer unit vector normal to an undeformed body, is the permutation symbol, and means the jump in any field quantity 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 such that the crack is placed on the -axis from to as shown in Figure 1, and assume that plane strain is perpendicular to the -axis. A uniform electric field is applied perpendicular to the crack surface. For convenience, all electric quantities outside the solid will be denoted by the superscript . The solution for the rigid body state is

(3.1)

The equations of motion are given by

(3.2)

where is the two-dimensional Laplace operator in the variables , , is the Poisson's ratio, is the shear wave velocity, and . The electric field equations for the perturbed state are

(3.3)

The electric field equations (3.3) are satisfied by introducing an electric potential such that

(3.4)

The displacement components can be written in terms of two scalar potentials and as

(3.5)

The equations of motion become

(3.6)

where is the compression wave velocity.

Let an incident plane harmonic compression wave (P-wave) be directed at an angle with the -axis so that

(3.7)

where is the amplitude of the incident P-wave, and is the circular frequency. The superscript stands for the incident component. Similarly, if an incident plane harmonic shear wave (SV-wave) impinges on the crack at an angle with -axis, then

(3.8)

where 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 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:

(3.9)

Mode II:

(3.10)

where the subscript and 2 correspond to the incident P- and SV-waves, , , , , and, are the compression and shear wave numbers, respectively, and .

## 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 and (2) antisymmetric with respect to .

### 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

(4.1)
(4.2)
(4.3)

where the subscript stands for the symmetric part. It can be shown that solutions , , , and of (3.4) and (3.6) for are

(4.4)
(4.5)
(4.6)

where , , and are unknown functions, and and are

(4.7)

The functions and should be restricted as

(4.8)

in the upper half-space , 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:

(4.9)

The boundary condition of (4.1) leads to the following relation between unknown functions:

(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:

(4.11)
(4.12)

in which and are known functions given by

(4.13)

and the original unknowns and are related to the new one through

(4.14)

The set of two simultaneous dual integral equations (4.11) and (4.12) may be solved by using a new function , and the result is

(4.15)

where is the zero-order Bessel function of the first kind, and and are

(4.16)

The function is governed by the following Fredholm integral equation of second kind:

(4.17)

where the kernel is given by

(4.18)
(4.19)
(4.20)

The kernel function (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

(4.21)

where the contours are defined in Figure 2, are, respectively, the zero-order Hankel functions of the first and second kinds, and

(4.22)

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

(4.23)

where

(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 -axis as in . Therefore for can be finally written as

(4.25)

where

(4.26)

The kernel is symmetric in , and the value of this kernel for is obtained by interchanging and in (4.25).

#### 4.1.2. Antisymmetric Solution for Mode I Crack

The boundary conditions for anti-symmetric scattered fields can be written as

(4.27)
(4.28)
(4.29)

where the subscript stands for the anti-symmetric part. The solutions , , and are

(4.30)
(4.31)
(4.32)

where , , , and are unknown functions. The displacements and stresses are obtained as

(4.33)
(4.34)
(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:

(4.36)

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

(4.37)
(4.38)

in which the original unknowns are related to the new one through

(4.39)

The unknowns and can be found by the same method of approach as in the symmetric case. The results are

(4.40)

where is the first-order Bessel function of the first kind, and in (4.40) is the solution of the following Fredholm integral equation of the second kind:

(4.41)

where

(4.42)

By using the contours of integration in Figure 2, the kernel for can be rewritten in the form

(4.43)

where is the first-order Hankel function of the first kind. The value of for is obtained by interchanging and in (4.43).

#### 4.1.3. Mode I Dynamic Singular Stresses Near the Crack Tip

The mode I dynamic electric stress intensity factor is

(4.44)

where

(4.45)

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

(4.46)
(4.47)
(4.48)

The electric stress intensity factor may be obtained as

(4.49)

The dynamic stress intensity factor can be found as

(4.50)

The dynamic electroelastic stress is given by

(4.51)

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

(4.52)
(4.53)

where and are the polar coordinates. Also, the singular parts of the displacements and electric fields near the crack tip are

(4.54)
(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

(4.56)
(4.57)
(4.58)

Replace the subscript by , , , , and by , , , and , respectively, in (4.30)â€“(4.35). The boundary condition of (4.56) leads to

(4.59)

Introducing the abbreviation

(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 :

(4.61)
(4.62)

where

(4.63)

The solution of (4.61) and (4.62) are obtained by using two new functions and , and the results are

(4.64)

where and are the solutions of the following Fredholm integral equations of the second kind:

(4.65)
(4.66)

The kernels are given by

(4.67)

where

(4.68)

and . The kernels are symmetric in and .

#### 4.2.2. Antisymmetric Solution for Mode II Crack

The boundary conditions for anti-symmetric scattered fields are

(4.69)
(4.70)
(4.71)

Let replace the subscript by , , , , and by , , and in (4.4)â€“(4.6). The boundary condition of (4.69) leads to

(4.72)

Introducing the abbreviation

(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:

(4.74)
(4.75)

Equations (4.74) and (4.75) yield the solutions

(4.76)

and are the solutions of the following Fredholm integral equations of the second kind:

(4.77)
(4.78)

where

(4.79)

and are symmetric in and .

#### 4.2.3. Mode II Dynamic Singular Stresses Near the Crack Tip

The dynamic stress intensity factor is obtained as

(4.80)

where

(4.81)

The singular parts of the dynamic local stresses and Maxwell stresses near the crack tip can be derived as follows:

(4.82)

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

(4.83)

where

(4.84)

## 5. Dynamic Energy Release Rate

The dynamic energy release rate is obtained as

(5.1)

where is the region with the contour . 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

(5.2)

where

(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 can be found as .

Figure 3 exhibits the variation of the normalized mode I dynamic stress intensity factor against the normalized frequency subjected to -waves for the normalized electric field and the angle of incidence . The dynamic stress intensity factor drops rapidly beyond the first maximum and exhibits oscillations of approximately constant period as increases. The peak value of under is 1.364. Also, the peak values of under are 1.522, 2.416, 3.310 for , respectively. As ,â€‰â€‰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 , where is the static energy release rate. The peak values of under for are 1.861, 2.361, 5.838, 10.96, respectively. Figure 5 shows the normalized mode I dynamic stress intensity factor versus subjected to P-waves for and . The peak values of under are 1.078, 1.198 for , respectively. Figure 6 shows the normalized mode II dynamic stress intensity factor versus subjected to -waves for and . 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 against the angle of incidence subjected to -waves for and . The mode I dynamic stress intensity factors for and 0.8 attain its maximum values at an incident angle of approximately .

Figure 8 shows the variation of the normalized mode II dynamic stress intensity factor versus subjected to SV-waves for and . 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 against subjected to SV-waves for and . Similar trend to the case under P-waves is observed.

## 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.

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Correspondence to Yasuhide Shindo.

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Shindo, Y., Narita, F. Electroelastic Wave Scattering in a Cracked Dielectric Polymer under a Uniform Electric Field. Bound Value Probl 2009, 949124 (2009). https://doi.org/10.1155/2009/949124