Energy and volume changes due to the formation of a circular inhomogeneity in a residual deviatoric stress field

Abstract

An inclusion of purely dilatational eigenstrain in an infinitely extended isotropic elastic matrix, independently of its shape, causes a deviatoric stress field around it. The present paper analyses the energy and volume changes due to the formation of a circular inhomogeneity in a deviatoric stress field coming from a circular inclusion of dilatational eigenstrain. It is found that the elastic stress inside the inhomogeneity remains deviatoric and the inhomogeneity formation does not change the volume of the inclusion-matrix system; it is argued that the same occurs for any inclusion shape and non-uniform eigenstrain. The elastic energy changes occurring in the domains occupied by matrix, inhomogeneity, and inclusion are calculated, and its dependence on the elastic properties and geometrical parameters of inhomogeneity and matrix is numerically investigated. Strengthening effects of the matrix-inhomogeneity system are examined by means of the energy force and expanding moment acting on the inhomogeneity.

APPENDIX

Basic relationships

Muskelishvili’s relationships are [5]:

$$\begin{aligned} \sigma _{rr}^k +\sigma _{\theta \theta }^k= & {} 2[{\varphi }'_k (z)+\overline{{\varphi }'_k (z)} ], \end{aligned}$$

(A.1)

$$\begin{aligned} \sigma _{rr}^k -i\sigma _{r\theta }^k= & {} {\varphi }'_k (z)+\overline{{\varphi }'_k (z)} -[\bar{{z}}{\varphi }''_k (z)+{\psi }'_k (z)]e^{2i\theta },\end{aligned}$$

(A.2)

$$\begin{aligned} 2\mu _k (u_r^k +iu_\theta ^k )e^{i\theta }= & {} \kappa \varphi _k (z)-z\overline{{\varphi }'_k (z)} +\overline{\psi _k (z)} . \end{aligned}$$

(A.3)

The complex potentials of the inhomogeneity and matrix in terms of their two-phase potentials $\varphi _0 (z)$, $\psi _0 (z)$ [23] are:

$$\begin{aligned} \varphi _I (z)= & {} (1+\Lambda )\varphi _0 (z), \end{aligned}$$

(A.4)

$$\begin{aligned} \psi _I (z)= & {} (1+\Pi )\psi _0 (z), \end{aligned}$$

(A.5)

and

$$\begin{aligned} \varphi _M (z)= & {} \varphi _0 (z)-\Omega z\bar{{{\varphi }'}}_0 \left( {\frac{R_I^{2}}{z}} \right) +\Pi \bar{{\psi }}_0 \left( {\frac{R_I^{2}}{z}} \right) , \end{aligned}$$

(A.6)

$$\begin{aligned} \psi _M (z)= & {} \psi _0 (z)+\Lambda \bar{{\varphi }}_0 \left( {\frac{R_I^{2}}{z}} \right) +(\Lambda +\Omega )\frac{R_I^{2}}{z}{\varphi }'_0 (z) \nonumber \\&+\,\frac{R_I^{2}}{z}\frac{\hbox {d}}{\hbox {d}z}\left[ {\Omega z\bar{{{\varphi }'}}_0 \left( {\frac{R_I^{2}}{z}} \right) -\Pi \bar{{\psi }}_0 \left( {\frac{R_I^{2}}{z}} \right) } \right] \end{aligned}$$

(A.7)

where

$$\begin{aligned} \Lambda =\frac{\mu _I \kappa _M -\mu _M \kappa _I }{\mu _I +\mu _M \kappa _I },\quad \Pi =\frac{\mu _I -\mu _M }{\mu _M +\mu _I \kappa _M },\quad \Omega =-\frac{\mu _I -\mu _M }{\mu _I +\mu _M \kappa _I } \end{aligned}$$

(A.8)

are two-phase parameters alternative of those of Dundurs [27].