Uniform fields inside two non-elliptical inclusions

Abstract

The problem of two non-elliptical inclusions with internal uniform fields embedded in an infinite matrix, subjected at infinity to a uniform stress field, is discussed in detail by means of the conformal mapping technique. The introduced conformal mapping function can map the matrix region (excluding the two inclusions) onto an annulus. The problem is completely solved for anti-plane isotropic elasticity, anti-plane piezoelectricity, anti-plane anisotropic elasticity, plane elasticity and finite plane elasticity. The correctness of the solution is verified by comparison with existing solutions and by checking an extreme situation. Our results indicate that it is permissible for the two inclusions to have different material properties and different shapes. Finally, two interesting applications of the obtained results are given, and we find that when the two inclusions have different material properties, the elastic polarization tensor associated with the two non-elliptical inclusions does not lie on the lower Hashin–Shtrikman bound.

Uniform stress fields inside multiple inclusions in an elastic infinite plane under plane deformation

“…It is also shown in figure 2d that the uniform internal hydrostatic stress fields inside two first-category inclusions with different shear moduli can be exactly identical, and in figure 3b that the uniform internal stress fields inside multiple inclusions of the second category with the same Poisson's ratio can be different from each other. Particularly, figure 3b,d indicates that the shear stresses inside any two inclusions of multiple inclusions of the second category need not be the same, which extends the scope of the validity of a conclusion made in [23] (see after equation (77) of [23]). Shown in figure 4 are multiple inclusions of the mixed first and second categories, with various prescribed uniform internal stress fields inside the inclusions of the second category, under various remote loadings.…”

“…As the first two examples, figure 2 shows various shapes of multiple inclusions of the first category under various uniform remote loadings, and figure 3 shows various shapes of multiple inclusions of the second category which achieve various prescribed uniform internal stress fields under given remote loadings. 3 of [23]. Here, it should be stressed that although the inclusion shapes shown in fig.…”

“…Here, it should be stressed that although the inclusion shapes shown in fig. 1 of [7] and figs 2 and 3 of [23] are obtained in anti-plane shear, the methods of [7] and [23] suggest that they could also achieve uniform internal stress fields in plane stress/strain for specific elastic constants and remote in-plane loadings. Our results shown in figure 2 indicate that any admissible remote loading with |g 1 | < 1 for a single elliptical inclusion of the first category is admissible for multiple inclusions of the first category, or, more precisely, any remote loading will be admissible for multiple inclusions of the first category if all (not some) of the corresponding complex parameters g i (i = 1 .…”

“…According to equations (3.5) and (3.7), the internal stress fields (Γ i , Γ i ) inside such multiple inclusions of the second category should be prescribed as ( 6) which is equivalent to equation (75) of [23]. Here, equation (4.6) gives the conditions on the uniform internal stress fields which guarantee the existence of such inclusions of the second category whose shapes are independent of the remote loading for any given elastic constants κ and κ i (i = 1 .…”

Non-circular nano-inclusions with interface effects that achieve uniform internal strain fields in an elastic plane under anti-plane shear

“…Therefore, the present work aims to examine the existence of non-circular nano-inclusions with interface effects which achieve uniform internal strain fields in an elastic infinite plane subjected to uniform remote anti-plane shear loadings. It should be mentioned that the special shapes of inclusions with uniform internal fields constructed in [4][5][6][7][8][9] are restricted to the ideal case when the inclusion-matrix interfaces are perfectly bonded, whereas our present work focuses on the construction of nano-inclusions with uniform internal fields in the presence of imperfect interfaces incorporating interface tension and energy.…”