Issue Section:
Discussions
Keywords:
elasticity, anisotropic media, errors
Topics:
Anisotropy, Elasticity, Errors

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 wαξα=1,2,3⁠, to three different points ξα 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 Γαα=1,2,3 (that is Γα defined in 1), one can construct an auxiliary function Dαz which satisfies the condition  
z¯=Dαz,z∈Γα
(1)
and is analytic and single-valued in the exterior of the curve Γα, except at infinity where Dαz tends to a polynomial Pαz⁠. 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 Γα is mapped onto the exterior of the unit circle in the ξ-plane by a polynomial conformal mapping  
z=wαξ=λαξ+∑k=0Ncαkξ−k,α=1,2,3
(2)
where λα is a real number, cαk 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 wα−1z which is well defined on the exterior of the curve Γα and maps the exterior of the curve Γα (without any branch cut!) on to the exterior of the unit circle. Evidently, this means that the inverse mapping wα−1z is analytic, single-valued and nonzero in the exterior of the curve Γα. 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 wα−1z is treated as the known, and we need not discuss how to construct an explicit expression for the inverse mapping wα−1z 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 Γα 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 Dαz is given by  
Dαz=wα¯1wα−1z=λαwα−1z+∑k=0Ncαk¯[wα−1z]k
(3)
where wα−1z is the (unique) inverse mapping of the polynomial mapping (2). First, Dαz given by (3) meets the condition (1) on the curve Γα. Second, because wα−1z is analytic, single-valued and nonzero outside the curve Γα,1/wα−1z and [wα−1z]k(k is any integer not larger than N) are analytic and single-valued outside the curve Γα. Thus, Dαz given by the right-hand side of (3) is obviously analytic and single-valued in the exterior of the curve Γα, except at infinity where Dαz tends to a polynomial of degree N. Therefore, the auxiliary function Dαz complying with the conditions (1) can be constructed by (3) in terms of the associated polynomial mapping which maps the exterior to the curve Γα 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 Γα 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.

1.
Ting
,
T. C. T.
,
2000
, “
Common Errors on Mapping of Nonelliptic Curves in Anisotropic Elasticity
,”
ASME J. Appl. Mech.
,
67
, pp.
655
–
657
.
2.
Ru, C. Q., 2001, “Analytic Solution for an Inclusion of Arbitrary Shape in an Anisotropic Plane or Half-Plane” (submitted for publication).
3.
Ru
,
C. Q.
,
2000
, “
Eshelby’s Problem for Two-Dimensional Piezoelectric Inclusions of Arbitrary Shape
,”
Proc. R. Soc. London, Ser. A
,
A456
, pp.
1051
–
1068
.
4.
Ru
,
C. Q.
,
1999
, “
Analytic Solution for Eshelby’s Problem of an Inclusion of Arbitrary Shape in a Plane or Half-Plane
,”
ASME J. Appl. Mech.
,
66
, pp.
315
–
322
.
5.
Ivanov, V. I., and Trubetakov, M. K., 1995, Handbook of Conformal Mapping With Computer-Aided Visualization, CRC Press, Boca Raton, FL.
Copyright © 2001
 by ASME