Ellipsoidal Domains: Piecewise Nonuniform and Impotent Eigenstrain Fields

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.