Z. Angew. Math. Phys. (2018) 69:5
2017 Springer International Publishing AG,
part of Springer Nature
published online December 11, 2017
Zeitschrift f¨ur angewandte
Mathematik und Physik ZAMP
A concentrated couple near two non-elliptical inclusions with internal uniform
Xu Wang, Liang Chen and Peter Schiavone
Abstract. We employ conformal mapping techniques to study the existence of internal uniform hydrostatic stresses inside
two non-elliptical inclusions when the surrounding matrix is simultaneously subjected to a concentrated couple and re-
mote uniform in-plane stresses. The unknown complex coeﬃcients appearing in the corresponding mapping function can
be determined analytically for a given pair of loading, one material and three geometric parameters. This allows us to
subsequently identify the shapes of the two inclusions. Our analysis further reveals that the shapes of the inclusions depend
on the concentrated couple, whereas the corresponding internal uniform hydrostatic stresses do not.
Mathematics Subject Classiﬁcation. 35Q74, 74B05, 74G70, 74G75, 30E25.
Keywords. Two non-elliptical inclusions, Concentrated couple, Uniform hydrostatic stresses, Conformal mapping.
The study of problems involving elastic inclusions has a long history (see the earlier review by and
the recent review by ). In particular, the problem of determining nonstandard shapes of multiple
interacting elastic inclusions (or inhomogeneities) which enclose uniform stress distributions continues
to attract considerable interest in the literature (see, for example, [3,4,6,7,14,15]). In particular, it was
recently demonstrated by Wang and Schiavone  that two non-elliptical inclusions interacting with a
single or even multiple screw dislocations continue to maintain a state of internal uniform stress when
the matrix is subjected to remote uniform anti-plane stresses. Most recently, Wang and Schiavone 
have shown that an internal uniform hydrostatic stress ﬁeld can be achieved inside a single non-elliptical
inclusion when the matrix is simultaneously subjected to remote uniform in-plane stresses and a concen-
trated couple (or point moment in the context of ). The importance of the analysis of elastic ﬁelds
subjected to pointwise singularities such as a concentrated couple (or point moment) is well documented.
These singular ﬁelds are important not only because of their physical signiﬁcance within micromechanics
(see, for example, ) but also because they often form the basis for fundamental solutions used in, for
example, the boundary integral equation method.
In this paper, we continue this interesting yet challenging line of investigation and examine the ex-
istence of internal uniform hydrostatic stresses inside two non-elliptical inclusions when the matrix is
subjected to remote uniform in-plane stresses and a randomly located concentrated couple. The analysis
involves the introduction of a conformal mapping function which maps the region occupied by the matrix
in the physical z-plane onto an annulus in the mapped ξ-plane. Two additional ﬁrst-order poles located
outside the annulus are introduced into the mapping function to account for the existence of the con-
centrated couple acting in the matrix. Once the single material parameter, two loading parameters and
three geometric parameters are identiﬁed, all of the unknown complex constants appearing in the map-
ping function can be uniquely determined. The results indicate that: (i) the internal uniform hydrostatic