Elastic and piezoelectric fields due to polyhedral inclusions

Abstract

We derive explicit, closed-form expressions describing elastic and piezoelectric deformations due to polyhedral inclusions in uniform half-space and bi-materials. Our analysis is based on the linear elasticity theory and Green's function method. The method involves evaluation of volume and surface integrals of harmonic and bi-harmonic potentials. In case of polyhedra, such integrals are expressed through algebraic functions. Our results generalize numerous studies on this subject, and they allow to obtain fully analytical solutions for a number of physical and engineering problems. In the limiting case of an infinite space, our relations have an essentially more compact form, than relations obtained by other authors. We present solutions to classical Mindlin and Cherruti problems. We describe the elastic relaxation of a misfitting polygonal quantum dot in bi-materials assuming isotropic and vertically isotropic properties. It is explained how to analyze nonhydrostatic and non-uniform inclusions. We also study piezoelectric fields induced by inclusions in materials with cubic and hexagonal lattices. Among other results, we have found that a cubic inclusion in an isotropic material reproduces fields of quantum dots in GaAs (0, 0, 1) and GaAs (1, 1, 1) depending on the orientation of the cube. This suggests that one can qualitatively model crystals with different lattices by choosing an appropriate inclusion shape.

Analytical solutions for elastic fields caused by eigenstrains in two joined and perfectly bonded half-spaces and related problems

“…Elastic fields caused by ellipsoidal inclusions with eigenstrains in a full space were investigated by many researchers (Mura, 1993;Zhou et al, 2013;Juan et al, 2014) following the direction of Eshelby (1957Eshelby ( , 1959. Other shapes of inclusions were also studied, such as cuboid inclusions (Chiu, 1978), polyhedral inclusions (Nozaki and Taya, 2001;Kuvshinov, 2008;Gao and Liu, 2012). For the half-space problems, the analytical results were derived for the problems of spherical thermal inclusions (Mindlin and Cheng, 1950b), ellipsoidal inclusions (Seo and Mura, 1979), cylindrical inclusions (Wu and Du, 1996), and cuboidal inclusions (Chiu, 1978).…”

Anisotropic phase field solution for morphological evolution and migration of inclusions in piezoelectric films

“…[16][17][18][19][20][21][22][23][24]. By means of an analytical method, Wang and Zhong investigated the problem of a conducting arc crack between a circular piezoelectric inclusion and an infinite piezoelectric matrix [16].…”

“…By utilizing the Euler-Bernoulli beam model and Rayleigh-Rita approximation technique, Della and Shu gave a mathematical model for vibration of beams with piezoelectric inclusions [20]. Kuvshinov [21] presented explicit and closed-form solution for electroelastic deformations due to polyhedral inclusion in uniform half-space and bi-materials utilizing Green function method. Fakri and Azrar [22] predicted the electroelastic and thermal responses of piezoelectric composites with and without voids.…”

Disturbed elastic fields in a circular 2D finite domain containing a circular inhomogeneity and a finite interfacial zone

“…To overcome this obstacle, Nozaki and Taya [7,8] proposed an approximate approach to tackle the polygonal (or polyhedral) inhomogeneous inclusion in a 2D (or 3D) infinite domain. Alternatively, methods employing Green's function combined with surface/volume integral evaluation of harmonic and bi-harmonic potentials [9] or dislocation loop [10] were proposed for inhomogeneous inclusions in an infinite domain. In these derivations, the size of the interfacial zone between the inclusion and matrix is deemed negligible, and the RVE is modeled as a 2-phase composite solid.…”