Professor Ting’s paper (1) clearly clarifies several simple but important concepts on conformal mapping techniques applied to anisotropic plane elasticity. Here, I would like to add my own comments on these interesting issues.
(1) First, it should be stated that conformal mapping techniques, combined with the Stroh’s method, have been successfully applied in some important cases to anisotropic elasticity with nonelliptical curves. An example is the Eshelby’s problem for an inclusion of arbitrary shape in an anisotropic medium (2), or in a piezoelectric medium 3, of the same material constants. As stated by Prof. Ting in 1, and also by some other authors elsewhere, because a point z on Γ will be transformed, under three different mappings , to three different points on the unit circle in ζ-plane, the transformed boundary conditions on the unit circle in the ζ-plane will contain three unknown Stroh’s functions which take values at three different points. Therefore, unless the boundary conditions are decoupled for the three Stroh’s functions, one cannot solve the transformed boundary value problem in the ζ-plane. The key fact associated with the problem studied in 2 is that the three interface conditions (in complex form) for an arbitrarily shaped inclusion, surrounded by an anisotropic medium of the same material constants, can be written in a decoupled form in which the three unknown Stroh’s functions are completely decoupled to each other. It is this fact that allows one to apply conformal mapping techniques to each of the three Stroh’s functions and the associated curve independently of the other two. For a similar result for piezoelectric materials, see 3.
(2) Finally, as stated in 1, the mapping (15) of 1, although provides a one-to-one mapping for the boundaries, does not always offer a one-to-one mapping for the exteriors of the boundaries. Regarding this issue, as stated in 2,3, the boundary correspondence principle of conformal mappings for exterior domains (5) can be used to identify the conditions under which a one-to-one mapping for the boundaries automatically offers a one-to-one mapping for the exteriors. For instance, for an elliptical boundary Γ, because the right-hand side of (15) of 1 is analytic outside the unit circle and has a simple pole (of degree one) at infinity in the ξ-plane, it follows from the boundary correspondence principle 5 that the expression (15) of 1 provides a one-to-one conformal mapping between the exterior of the curve and the exterior of the unit circle in the ξ-plane, not any pointwise verification is needed. I believe that this comment offers a valuable insight to this interesting issue.