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Ellipsoidal Domains: Piecewise Nonuniform and Impotent Eigenstrain Fields

Authors:

Institution:

(1) Department of Civil Engineering, Center of Excellence in Structures and Earthquake Engineering, Sharif University of Technology, P.O. Box 11365-9313, Tehran, Iran;(2) Institute for Nanoscience and Nanotechnology, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran

Abstract:

In association with multi-inhomogeneity problems, a special class of eigenstrains is discovered to give rise to disturbance stresses of interesting nature. Some previously unnoticed properties of Eshelby’s tensors prove useful in this accomplishment. Consider the set of nested similar ellipsoidal domains {Ω₁, Ω₂,⋯,Ω_N+1}, which are embedded in an infinite isotropic medium. Suppose that

$\Omega_{t}=\{ {\textbf{x}} \mid {\textbf{x}} \in \mathbb{R}^{3} , \quad \sum_{p=1}^3\frac{x_{p}^2}{a_{p}^{2}}\leqslant\xi_{t}^2 \},$

in which $0\leqslant\xi_{1} < \xi_{2} < \cdots < \xi_{N+1}$ and ξ _t a _p, p=1,2,3 are the principal half axes of Ω_t. Suppose, the distribution of eigenstrain, ε _ij ^*(x) over the regions Γ _t=Ω_t+1−Ω_t , t=1,2,⋯,N can be expressed as

$\epsilon_{ij}^{*} \left( {\textbf{x}} \right)=\begin{cases} f_{ijkl \cdots m }^{(t)}\biggl( \sum\limits_{p=1}^3\dfrac{ x_{p}^2 }{a_{p}^{2}} \biggr) x_{k}x_{l} \cdots x_{m}, \qquad {\textbf{x}} \in \Gamma_{t},\\ \quad 0, \qquad\qquad\qquad\qquad\qquad\ \qquad {\textbf{x}} \in \Omega_1\bigcup \left(\mathbb{R}^3-\Omega_{N+1}\right), \end{cases}$

(‡)

where x _k x _l ⋯x _m is of order n, and f _{ijkl ⋯m} ^(t) represents 3N(n+2)(n+1) different piecewise continuous functions whose arguments are ∑_p=1 ³ x _p ² /a _p ². The nature of the disturbance stresses due to various classes of the piecewise nonuniform distribution of eigenstrains, obtained via superpositions of Eq. (‡) is predicted and an infinite number of impotent eigenstrains are introduced. The present theory not only provides a general framework for handling a broad range of nonuniform distribution of eigenstrains exactly, but also has great implications in employing the equivalent inclusion method (EIM) to study the behavior of composites with functionally graded reinforcements. The paper is dedicated to professor Toshio Mura.

Keywords:

ellipsoidal domain similar ellipsoids nonuniform eigenstrain impotent eigenstrain Eshelby’ s tensor 3D elastic fields exact solution

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