Ellipsoidal Domain with Piecewise Nonuniform Eigenstrain Field in One of Joined Isotropic Half-Spaces

Abstract

Consider an arbitrarily oriented ellipsoidal domain near the interface of an isotropic bimaterial space. It is assumed that a general class of piecewise nonuniform dilatational eigenstrain field is distributed within the ellipsoidal domain. Two theorems relevant to prediction of the nature of the induced displacement field for the interior and exterior points of the ellipsoidal domain are stated and proved. As a resultant the exact analytical expression of the elastic fields are obtained rigorously. In this work a new Eshelby-like tensor, A is introduced. In particular, the closed-form expressions for A associated with the interior points of spherical and cylindrical inclusion are derived. The stress field is presented for a single ellipsoidal inclusion which undergoes a Gaussian distribution of eigenstrain field and one of the principal axes of the domain is perpendicular to the interface. For the limiting case of spherical inclusion the closed-form solution is obtained and the associated strain energy is discussed. For further demonstration, two examples of two concentric spheres and three concentric cylinders with eigenstrain field distributions which are descriptive of the general class of functions defined in this paper. The effect of some parameters such as distance between the inclusion and the interface, and the ratio of the shear moduli of the two media on the induced elastic fields are examined.

The Fundamental Formulation for Inhomogeneous Inclusion Problems with the Equivalent Eigenstrain Principle

“…Once the equivalent eigenstrain (ε 0 kl ) is fixed, the inclusion problem will be completely solved. A couple of related numerical methods have been developed in the literature, which can be directly adopted to numerically solve Equation (43). They are briefly introduced as follows:…”

“…The unknown equivalent eigenstrain in each inclusion can be expressed into a Taylor series with respect to the local origin of the inclusion, and the singular integrals will automatically become an integrable one [50]. The coefficients of the series are then determined by inserting them into the governing equation, Equation (43), which is also called the "consistency condition"; (b) The discretized element method. To find a more accurate numerical solution, the inhomogeneous inclusions are discretized into small elements, each of which is treated as a homogenous inclusion with an initial eigenstrain plus an unknown equivalent eigenstrain, according to the equivalent inclusion method (see, e.g., [56,58,59,62]), and the equivalent eigenstrain in each element should satisfy the discretized consistency condition Equation ( 43); (c) The Fourier transform method (or the numerical FFT method).…”

“…The basic idea is to replace the inhomogeneous inclusion with an equivalent homogeneous one, which has a new equivalent eigenstrain distribution. Then, the homogenous inclusion can be easily solved by using Green's function method (see, e.g., [13,[38][39][40][41][42][43][44][45][46]). This treatment may largely simplify the difficulty of the original inhomogeneous inclusion problems, and, thus, the analytical solutions for a series of inhomogeneous inclusion problems with nonuniform eigenstrain distributions can be developed (see, e.g., [44,[47][48][49]).…”

Two bonded half‐planes with a parabolic inclusion

“…The Eshelby problem of an Eshelby inclusion undergoing uniform or nonuniform stress‐free eigenstrains has been thoroughly investigated by many authors (see, for example, [1–17]). The Eshelby problem arises mainly from thermal and intrinsic stresses caused by thermal and lattice mismatch between dissimilar materials.…”

Boundary effect on the elastic field of a semi-infinite solid containing inhomogeneities

“…However, when an inhomogeneity is subjected to a body force or a far-field stress, the mismatch between inhomogeneity and the matrix can be simulated by an appropriately chosen eigenstrain on the subdomain so that the elastic field can be obtained through solving the inclusion problem [25,40]. Notice that the inclusion problems for a semi-infinite domain or two half-infinite domains have been studied [36,37,41,42]. However, the inhomogeneity problem for a semi-infinite domain has not been solved in the literature yet.…”