Transfer matrix method for the solution of multiple elliptic layers with different elastic properties. Part I: infinite matrix case

This paper provides the transfer matrix method for the solution of multiple elliptic layers with different elastic properties. In the study, the medium is composed of an elliptic inclusion and many confocal elliptic layers. The conformal mapping and the continuation of analytic functions are used. In the mapping plane, the complex potentials in the inclusion and the individual layer are expressed in the form of Laurent series. The correct form of the complex potentials in the inclusion is addressed. From the continuation condition for traction and displacement along the interface, the relation between the coefficients in the Laurent series of complex potentials for two adjacent layers can be evaluated. This relation is expressed in a matrix form, and it is called the transfer matrix. Using the transfer matrixes successively and the traction condition at remote place, the problem is finally solved. Numerical examples for two cases, the two-phase case and the three-phase case, are presented.