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.
The second key result of (
2) is that for each of the three closed curves
(that is
defined in
1), one can construct an auxiliary function
which satisfies the condition
and is analytic and single-valued in the exterior of the curve
except at infinity where
tends to a polynomial
. As shown in
2 (and
3 for piezoelectric materials), with aid of these auxiliary functions, the techniques of analytic continuation can be applied to the inclusion of arbitrary shape to get an analytic solution for the Stroh’s functions.
The above key result (which has been questioned by someone!) can be shown, clearly and rigorously, as follows. Assume that the exterior of
is mapped onto the exterior of the unit circle in the ξ-plane by a polynomial conformal mapping
where
is a real number,
are some complex constants, and
N is a finite integer. It is emphasized that the definition of the conformal mapping (
2) implies that it has a unique inverse conformal mapping
which is well defined on the exterior of the curve
and maps the exterior of the curve
(without any branch cut!) on to the exterior of the unit circle. Evidently, this means that the inverse mapping
is analytic, single-valued and nonzero in the exterior of the curve
This is just part of the definition (
2)—not any further “proof” is needed. Here, similar to all other conformal mapping methods, the inverse mapping
is treated as the known, and we need not discuss how to construct an explicit expression for the inverse mapping
from the single-valued branches of the multivalued inverse function of (
2). In particular, all branch points of the inverse function of (
2) fall inside the interior of the curve
in the
z-plane, or inside the interior of the unit circle in the ξ-plane. For example, for a hypotrochoidal curve, it is readily seen from (A11) of
4 that all singularity points of the conformal mapping (at which the derivative of the mapping function vanishes, as described by the condition (3) or (6) of
1) fall inside the interior of the unit circle in the ξ-plane and thus do not trouble the single-valued inverse mapping for the exterior.
Based on these facts, it is easily verified that the desired function
is given by
where
is the (unique) inverse mapping of the polynomial mapping (
2). First,
given by (
3) meets the condition (
1) on the curve
Second, because
is analytic, single-valued and nonzero outside the curve
and
is any integer not larger than
N) are analytic and single-valued outside the curve
Thus,
given by the right-hand side of (
3) is obviously analytic and single-valued in the exterior of the curve
except at infinity where
tends to a polynomial of degree
N. Therefore, the auxiliary function
complying with the conditions (
1) can be constructed by (
3) in terms of the associated polynomial mapping which maps the exterior to the curve
onto the exterior of the unit circle. Similar auxiliary functions have been applied to isotropic elasticity
4 and piezoelectric materials
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.
