On Approximate Solutions for Two-Dimensional Thermoelastic Problems with a Nearly Circular Hole

ABSTRACTThe general approximate solutions for the two-dimensional thermoelastic problems with a nearly circular hole are provided in this study. Based on Stroh formalism and the method of conformal mapping, the boundary perturbation analysis is applied to solve the problems of a hole with arbitrary shape. The radius of the hole considered here is represented as a sum of a reference constant and a perturbation magnitude that is expanded into a Fourier series. In order to illustrate the applicability and efficiency of the present approach, special examples associated with polygonal hole problems are solved explicitly and discussed in detail. Since the general solutions have not been found in the literature, comparison is made with some special cases for which the analytical solutions exist, which shows that our proposed method is effective and general.

Rigid Stamp Indentation on a Curvilinear Hole Boundary of an Anisotropic Elastic Body

A general solution for the problems of rigid stamp indentation on a curvilinear hole boundary of an anisotropic elastic body is obtained by employing the Stroh formalism, the method of analytical continuation, and the technique of mapping a closed curve to a unit circle. With this general solution, two typical curvilinear holes are studied. One is an ellipse, the other is a polygon. Since the transformation functions used in our solutions are not always one to one, some of the solutions are not exact but only approximate. For example, the solutions to the problems of anisotropic plates containing an elliptic hole and isotropic plates containing a polygonal hole are exact, but the solutions to the problems of anisotropic plates containing a polygonal hole are only approximate. Because the solutions presented in this paper are new and no other analytical solution has been found in the literature, the correctness of the present results can only be checked analytically by their reduced forms such as those for isotropic media or those for the stress boundary value problems. To show the generality of our solutions and to see clearly the physical behavior of the indentation problems two numerical examples are given, and their related hoop stress and stress contour are also plotted.

The conductivity of a sheet containing a few polygonal holes and/or superconducting inclusions

I use conformal mapping techniques to determine the change in the conductivity of a sheet containing a few well-separated holes. The hole shapes studied are the equilateral triangle, square, pentagon and regular n -gons. I show that the conductivity can be written as σ / σ 0 = 1 – α n f + o ( f 2 ), where f is the area fraction of the inclusions and the coefficient α n = (tan (π/ n )/2π n ) Г 4 (1/ n )/Г 2 (2/ n ), which is 2.5811, 2.1884, 2.0878 for triangles, squares and pentagons, and tends to the circle limit of 2 as n →∞ . The coefficient α n is proportional to the induced dipole moment around the polygonal hole which can be found using an appropriate conformal mapping. I have also examined and compared the results for long thin needle-like holes in the shape of diamonds, rectangles and ellipses. In all cases the conductivity parallel to the needles has the limiting form σ / σ 0 = 1 – f , while for the perpendicular conductivity, I find that σ / σ 0 = 1 – n π a 2 , where 2 a is the length of the needle, and n is the number of needles per unit area. For thicker needles, the shape becomes important and I compare the results with recent analog experiments and computer simulations. Because of the reciprocity theorem, all the results found here apply equally well to superconducting inclusions.