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

Abstract

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 Green's 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 Eshelby's inclusion problem.

Counter-examples to the high-order version and strong version of the generalized Eshelby conjecture for anisotropic media

“…Let Γ(x − y):= − 1 4π|x−y| . Then owing to (28) 1 and (28) 3 , for any v od ∈ W 2,∞ loc (R 3 ), we can get…”

“…Therefore, we have proved that φ * satisfies all of the properties that an obstacle function φ must possess. Then for φ * , the over-determined problem (28) with φ replaced by φ * admits a corresponding solution…”

“…There are studies dealing with the non-ellipsoidal or non-elliptical inclusions like polygons [27,28,29] with polynomial eigenstrains prescribed, but the results only show that the considered nonellipsoidal or non-elliptical inclusions do not exhibit Eshelby's polynomial property, which does not lead to either falsification or substantiation of the high-order Eshelby conjecture. In this work, with the help of the variational method proposed by Liu [25], we firstly prove the existence of a threedimensional non-ellipsoidal inclusion that possesses Eshelby's polynomial property in linearly elastic anisotropic media of cubic, transversely isotropic, orthotropic, and monoclinic symmetries, when the eigenstrain is expressed in the form of a polynomial of degree two.…”

Analysis of Functionally Graded Quantum-Dot Systems with Graded Lattice Mismatch Strain

“…10 Exact closed-form solutions were developed for QDs of different shapes (spherical, cylindrical, ellipsoidal, pyramidal, and arbitrarily shaped polygonal) with graded eigenstrain in piezoelectric matrix (See Refs. [11][12][13] and the references therein). For example, exact closed-form solutions were derived for an arbitrarily shaped polygonal inclusion with any order of polynomial eigenstrains in an anisotropic magneto-electro-elastic full plane.…”

“…For example, exact closed-form solutions were derived for an arbitrarily shaped polygonal inclusion with any order of polynomial eigenstrains in an anisotropic magneto-electro-elastic full plane. 11 Solutions of linearly 12 and quadratically 13 graded eigenstrain in an anisotropic piezoelectric half plane were also developed. All the developed analytical models relied on two main assumptions:…”