Classical image treatment of a geometry composed of a circular conductor partially merged in a dielectric cylinder and related problems in electrostatics

Closed-form solutions for a mode III radial matrix crack penetrating a circular inhomogeneity

“…We find that the resulting boundary value problem in the new f-plane is very similar to the electrostatic problem recently discussed by Palaniappan [8] of a 2D snowman type of an object with a conducting cylinder partially protruded into a dielectric cylinder with a dielectric constant different from that of the surrounding host medium. It shall be mentioned that the traction-free boundary condition on the crack surface discussed here is of Neumann-type; while the zero-potential on the conductor surface discussed by Palaniappan [8] is of Dirichlet-type. In the next section we will derive closed form solutions in the f-plane by using the method of image (or continuously using analytic continuation on C a and C b ).…”

“…(2)) and the method of image [8], closed-form solutions are derived for the elastostatic problem of a mode-III radial matrix crack penetrating a circular elastic inhomogeneity. Detailed results are given for three cases: (1) the two-phase composite is subjected to remote uniform antiplane shearing; (2) a screw dislocation is located in the matrix; (3) the radial crack is a Zener-Stroh crack.…”

“…When the two-phase composite is only subjected to remote uniform shearing r 1 zy , by using the method of image [8] and also noticing the conformal mapping Eq. (2), the two analytic functions f 1 (f) defined in the inhomogeneity and f 2 (f) defined in the unbounded matrix can be expressed in closed-form as…”

“…By using the method of image [8], the two analytic functions f 1 (f) defined in the inhomogeneity and f 2 (f) defined in the unbounded matrix can be expressed in closed-form as…”

“…In addition the intersection angle of the two circles in the f-plane is p/2. Here it is stressed that the resulting boundary value problem in the new f-plane is similar to the electrostatic problem of a circular conductor partially merged in a dielectric cylinder with a dielectric constant different from that of the surrounding host medium, which was recently discussed by Palaniappan [8]. By using Kelvin's inverse (or method of image) [9][10][11][12] together with shift and reflection properties of harmonic functions, he analytically solved the exterior boundary value problem.…”