Application of Hankel Transform for Solving a Fracture Problem of a Cracked Piezoelectric Strip Under Thermal Loading

In this chapter, an example of the application of Hankel transform for solving a fracture problem will be explained. In discussing axisymmetric problems, it is advantageous to use polar coordinates, and the Hankel transform method is powerful to solve the general equations in polar coordinates. A brief account of the Hankel transform will be given. Here f is a function of r , its transform is indicated by a capital F , Jν is the ν th order Bessel function of the first kind, and the nature of the transformation either by a suffix or by a characteristic new variable s . It will be assumed without comment that the integrals in question exist, and that, if necessary, the functions and their derivatives tend to zero as the variable tends to infinity. The Hankel transform of order 1 / 2 ν > − , [ ( )] H f r ν or ( ) F s ν , of a function ( ) f r is defined as


Introduction
In this chapter, an example of the application of Hankel transform for solving a fracture problem will be explained.In discussing axisymmetric problems, it is advantageous to use polar coordinates, and the Hankel transform method is powerful to solve the general equations in polar coordinates.A brief account of the Hankel transform will be given.Here f is a function of r , its transform is indicated by a capital F , J ν is the ν th order Bessel function of the first kind, and the nature of the transformation either by a suffix or by a characteristic new variable s .It will be assumed without comment that the integrals in question exist, and that, if necessary, the functions and their derivatives tend to zero as the variable tends to infinity.The Hankel transform of order 1/2 The piezoelectric materials have attracted considerable attention recently.Owing to the coupling effect between the thermo-elastic and electric fields in piezoelectric materials, thermo-mechanical disturbances can be determined form measurement of the induced electric potential, and the ensuing response can be controlled through application of an appropriate electric field (Rao & Sunar, 1994).For successful and efficient utilization of www.intechopen.comFourier Transform -Materials Analysis 208 piezoelectric as sensors and actuators in intelligent systems, several researches on piezothermo-elastic behavior have been reported (Tauchert, 1992).
Moreover a better understanding of the mechanics of fracture in piezoelectric materials under thermal load conditions is needed for the requirements of reliability and lifetime of these systems.Using the Fourier transform, the present author studied the thermally induced fracture of a piezoelectric strip with a two-dimensional crack (Ueda, 2006a(Ueda, , 2006b)).
Here the mixed-mode thermo-electro-mechanical fracture problem for a piezoelectric material strip with a penny-shaped crack is considered.It is assumed that the strip is under the thermal loading.The crack faces are supposed to be insulated thermally and electrically.By using the Hankel transform (Sneddon & Lowengrub, 1969), the thermal and electromechanical problems are reduced to a singular integral equation and a system of singular integral equations (Erdogan & Wu, 1996), respectively, which are solved numerically (Sih, 1972).Numerical calculations are carried out, and detailed results are presented to illustrate the influence of the crack size and the crack location on the stress and electric displacement intensity factors.The temperature, stress and electric displacement distributions are also presented.T and 20 T are maintained over the stress-free boundaries.In the following, the subscripts rz θ , , will be used to refer to the direction of coordinates.The material properties, such as the elastic stiffness constants, the piezoelectric constants, the dielectric constants, the stress-temperature coefficients, the coefficients of heat conduction, and the pyroelectric constant, are denoted by kl c , kl e , kk ε ,
The constitutive equations for the elastic field are where where 2 rz κκ κ =/.
The boundary conditions can be written as for the electromechanical conditions.

Temperature field
For the problem considered here, it is convenient to represent the temperature as the sum of two functions. ( where (1) () Tz satisfies the following equation and boundary conditions: (1) 11 0 (1) 22 0 () , () It is easy to find from Eqs.( 12) and ( 13) that By applying the Hankel transform to Eq.( 14) (Sneddon & Lowengrub, 1969), we have where ()( 12) ij Dsi j ,=, are unknown functions to be solved and Taking the second boundary condition (15) into consideration, the problem may be reduced to a singular integral equation by defining the following new unknown function 0 () Making use of the first boundary condition (15) with Eqs.( 16), we have the following singular integral equation for the determination of the unknown function 0 () In Eq.( 21), the kernel functions (1)  0 () M tr , and ( 2 where K and E are complete elliptic integrals of the first and second kind, and Once 0 () Gt is obtained from Eq.( 21), the temperature field can be easily calculated as follows: where On the plane 0 z = , the temperatures (2) (0 ) ( 1 2 )

Thermally induced elastic and electric fields
The non-disturbed temperature filed (1) () Tz given by Eq.( 17) does not induce the stress and electric displacement components, which affect the singular field.Thus, we consider the elastic and electric fields due to the disturbed temperature distribution (2) as the sum of two functions, respectively. (1) (2) where (1)  Using the displacement potential function method (Ueda, 2006a), the particular solutions can be obtained as follows: where the constants (1) (1 21 2 6 ) kij pi j k , = , , = , ,..., are given in Appendix A. The general solutions are obtained by using the Hankel transform technique (Sneddon & Lowengrub, 1969): Similar to the temperature analysis, the problem may be reduced to a system of singular integral equations by taking the second boundary conditions ( 7)-( 9) into consideration and by defining the following new unknown functions () ( 123 )   l Gr l = ,, : Making use of the first boundary conditions ( 7)-( 9) with Eqs.(10), we have the following system of singular integral equations for the determination of the unknown functions ()( 123) { } ( In Eq.( 40), the functions 1 ()( 12 where the functions 1 () ds (1 26 ) j = , ,..., are also given in Appendix C.These components are superficial quantities and have no physical meaning in this analysis.However, they are equivalent to the crack face tractions in solving the crack problem by a proper superposition.
To solve the singular integral equations ( 21) and ( 35)-( 37) by using the Gauss-Jacobi integration formula (Sih, 1972), we introduce the following functions ()( 0123 Then the stress intensity factors I K , II K and the electric displacement intensity factor D K may be defined and evaluated as:

Numerical results and discussion
For the numerical calculations, the thermo-electro-elastic properties of the plate are assumed to be ones of cadmium selenide with the following properties (Ashida & Tauchert, 1998).
The values of the coefficients of heat conduction for cadmium selenide could not be found in the literature.Since the values of them for orthotropic Alumina (Al hh / on the stress and electric displacement intensity factors, the solutions of the system of the singular integral equations have been computed numerically.
In the first set of calculations, we consider the temperature field and the electro-elastic fields without crack.Figure 2 shows the normalized temperature 20 0  In the second set of calculations, we study the influence of the crack size on the stress and electric displacement intensity factors.Figures 4(a with increasing ch / .The value of I K for 1 075 hh / =. becomes negative so that the contact of the crack faces would occur.The results presented here without considering this effect may not be exact but would be more conservative.Since the contact of the crack faces will increase the friction between the faces and make thermo-electrical transfer across the crack faces easier, the stress and electric displacement intensity factors would be lowered by these two factors.
In the final set of calculations, we investigate the influence of the crack location on the intensity factors.Figure 5  Fig. 5 The effect of the crack location on the stress intensity factors I K , II K and the electric displacement intensity factor D K

Conclusion
An example of the application of Hankel transform for solving a mixed-mode thermoelectro-elastic fracture problem of a piezoelectric material strip with a parallel penny-shaped crack is explained.The effects of the crack size () ch / and the crack location 1 () hh / on the fracture behavior are analyzed.The following facts can be found from the numerical results.

Fig. 1 .
Fig. 1.Penny-shaped crack in a piezoelectric strip A penny-shaped crack of radius c is embedded in an infinite long piezoelectric strip of thickness 12 hh h =+ as shown in Figure 1.The crack is located parallel to the boundaries and at an arbitrary position in the strip, and the crack faces are supposed to be insulated thermally and electrically.The cylindrical coordinate system is denoted by () rz θ ,, with its origin at the center of the crack face and the plane r θ − along the crack plane, where z is the poling axis.It is assumed that uniform temperatures 10T and 20 T are maintained over = , ,..., are the unknown functions to be solved, and the constants ij γ and (2) (1 2 1 26 ) kij pi j k =,,, =,, . . ., are given in Appendix B. www.intechopen.com correspond to the stress and electric displacement components induced by the disturbed temperature field(2) / . is assumed.To examine the effects of the normalized crack size ch / and the normalized crack location 1 local temperature difference across the crack occurs at the center of the crack.

/Fig. 4 .
Fig. 3.The stress components 0 zz σ , 0 zr σ and the electric displacement component 0 z D on the r -axis without crack due to the temperature shown in Fig. 2 In the following, the superscripts (1) and (2) indicate the particular and general solutions of Eqs.(4).