Stress analysis of an orthotropic elastic infinite plane with a hole

A general solution (stress functions) is derived for an orthotropic elastic plane problem using a mapping function to solve arbitrary configurations. Lekhnitskii formulism using complex variable is used. A problem of an infinite orthotropic elastic plane with a hole is solved. The boundary condition is represented by two complex variables. This makes it difficult to solve the orthotropic elastic plane problem. The problem is solved by overcoming this difficulty. Two methods of Cauchy integral and Riemann-Hilbert problem are applied for the analysis. It is confirmed that the same exact stress functions are finally obtained by these methods and are represented by an irrational mapping function as a closed form. These stress functions are the general solution for the holed problem. The Mittag-Leffler Theorem is used for the analysis. Therefore, there are two methods to solve an external boundary value problem. Stress components are represented by one complex variable. Therefore, it is not difficult to calculate the stress components. Arbitrarily shaped hole problem can be solved by changing the mapping function in the stress functions. The stress distributions for an infinite plane with a square hole subjected to uniform tension for Case I and III problems are shown. Symmetry of the stress distribution is lost for a Case III problem. A problem of a half plane with an edge crack is solved as Riemann-Hilbert problem. It is confirmed that the solution coincide with that obtained by Cauchy integral method.