W.-N. Zou

Solutions to Eshelby's problems of non-elliptical thermal inclusions and cylindrical elastic inclusions of non-elliptical cross section

Eshelby’s inclusion problem is solved for non-elliptical inclusions in the context of two-dimensional thermal conduction and for cylindrical inclusions of non-elliptical cross section within the framework of generalized plane elasticity. First, we consider a two-dimensional infinite isotropic or anisotropic homogeneous medium with a non-elliptical inclusion subjected to a prescribed uniform heat flux-free temperature gradient. Eshelby’s conduction tensor field and its area average are first expressed compactly in terms of two boundary integrals avoiding the usual singularity and then specified analytically for arbitrary polygonal inclusions and for inclusions characterized by the finite Laurent series. Next, we are interested in a three-dimensional infinite isotropic or transversely isotropic homogeneous medium with a cylindrical inclusion of a non-elliptical cross section that undergoes uniform generalized plane eigenstrains. The solution to this problem is obtained by decomposing a generalized plane eigenstrain tensor into a plane strain part and an anti-plane strain part, exploiting the mathematical similarity between two-dimensional thermal conduction and anti-plane elasticity, and combining the relevant results of Zou et al. (Zou et al. 2010 J. Mech. Phys. Solids 58, 346–372. (doi:10.1016/j.jmps.2009.11.008)) with those derived in the present work for Eshelby’s conduction tensor field and its area average.

Eshelby's problem of polygonal inclusions with polynomial eigenstrains in an anisotropic magneto-electro-elastic full plane

This paper presents a closed-form solution for the arbitrary polygonal inclusion problem with polynomial eigenstrains of arbitrary order in an anisotropic magneto-electro-elastic full plane. The additional displacements or eigendisplacements, instead of the eigenstrains, are assumed to be a polynomial with general terms of order M+N. By virtue of the extended Stroh formulism, the induced fields are expressed in terms of a group of basic functions which involve boundary integrals of the inclusion domain. For the special case of polygonal inclusions, the boundary integrals are carried out explicitly, and their averages over the inclusion are also obtained. The induced fields under quadratic eigenstrains are mostly analysed in terms of figures and tables, as well as those under the linear and cubic eigenstrains. The connection between the present solution and the solution via the Greens function method is established and numerically verified. The singularity at the vertices of the arbitrary polygon is further analysed via the basic functions. The general solution and the numerical results for the constant, linear, quadratic and cubic eigenstrains presented in this paper enable us to investigate the features of the inclusion and inhomogeneity problem concerning polynomial eigenstrains in semiconductors and advanced composites, while the results can further serve as benchmarks for future analyses of Eshelbys inclusion problem.

Symmetry types of the piezoelectric tensor and their identification

The third-order linear piezoelectricity tensor seems to be simpler than the fourth-order linear elasticity one, yet its total number of symmetry types is larger than the latter and the exact number is still inconclusive. In this paper, by means of the irreducible decomposition of the linear piezoelectricity tensor and the multipole representation of the corresponding four deviators, we conclude that there are 15 irreducible piezoelectric symmetry types, and thus further establish their characteristic web tree. By virtue of the notion of mirror symmetry and antisymmetry, we define three indicators with respect to two Euler angles and plot them on a unit disk in order to identify the symmetry type of a linear piezoelectricity tensor measured in an arbitrarily oriented coordinate system. Furthermore, an analytic procedure based on the solved axis-direction sets is also proposed to precisely determine the symmetry type of a linear piezoelectricity tensor and to trace the rotation transformation back to its natural coordinate system.

Thermal inclusions inside a bounded medium

In the context of thermal conduction taken as a prototype of numerous transport phenomena, a general method is elaborated to study Eshelbys problem of inclusions inside a bounded homogeneous anisotropic medium. This method consists in: (i) recasting by a linear transformation the initial problem into Eshelbys problem of the transformed inclusion inside the transformed finite isotropic medium and (ii) decomposing Eshelbys problem of a thermal inclusion embedded in a finite isotropic medium into the sub-problem of the same inclusion inside the associated infinite medium and the sub-problem of the finite ambient isotropic medium including no inclusion but undergoing appropriate compensating boundary conditions. The general method is applied in the two-dimensional situation and the corresponding temperature field and Eshelbys conduction tensor are explicitly expressed in terms of some curvilinear complex integrals for the Dirichlet and Neumann boundary conditions. Thus, the difficulties owing to the unavailability or non-existence of Greens function are overcome. The general results in the two-dimensional case are finally specified and illustrated by considering a finite circular medium with circular or polygonal inclusions.