Electroelastic Wave Scattering in a Cracked Dielectric Polymer under a Uniform Electric Field
© Y. Shindo and F. Narita. 2009
Received: 25 April 2009
Accepted: 18 May 2009
Published: 16 June 2009
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.
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  considered the isotropic elastic dielectric material and obtained the form of the constitutive relations for the stress and electric fields. Kurlandzka  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  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:
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
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
where is the Maxwell stress tensor, and are the Lamé 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
3. Problem Statement
The equations of motion are given by
The electric field equations (3.3) are satisfied by introducing an electric potential such that
The equations of motion become
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
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
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 satisfaction of the two mixed boundary conditions (4.2) and (4.3) leads to two simultaneous dual integral equations of the following form:
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
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
The integrands in (4.21) satisfy Jordan's lemma on the infinite quarter circles, so that,
4.1.2. Antisymmetric Solution for Mode I Crack
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:
The boundary conditions in (4.28) and (4.29) lead to two simultaneous dual integral equations of the following form:
By using the contours of integration in Figure 2, the kernel for can be rewritten in the form
4.1.3. Mode I Dynamic Singular Stresses Near the Crack Tip
Next, we examine the static electroelastric crack problem. The boundary conditions may be written as
The dynamic electroelastic stress is given by
The singular parts of the dynamic local stresses and Mexwell stresses near the crack tip can be expressed as
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
Introducing the abbreviation
The kernels are given by
4.2.2. Antisymmetric Solution for Mode II Crack
Introducing the abbreviation
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:
Equations (4.74) and (4.75) yield the solutions
4.2.3. Mode II Dynamic Singular Stresses Near the Crack Tip
The singular parts of the displacements and electric fields near the crack tip can be expressed as
5. Dynamic Energy Release Rate
where is the region with the contour . This expression may be thought of as an extension to the J-integral given in . If all the electrical field quantities are made to vanish, then (5.1) reduces to the dynamic energy release rate for the elastic materials . Writing the dynamic energy release rate expression in terms of the mode I dynamic stress intensity factor, there results
6. Results and Discussion
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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