Eshebly tensors for a finite spherical domain with an axisymmetric inclusion

Eshelby's problem of a spherical inclusion eccentrically embedded in a finite spherical body

“…β 0 (δ ik t l t j + δ il t k t j + δ jk t l t i + δ jl t k t i ), (3.8) where I ijkl = 1 2 (δ ik δ jl + δ il δ jk ) is the fourth-order unit tensor. It can be proved that (3.8) coincides with the results in Li et al [8], except for a sign misprint of the second term, as pointed out by Mejak [14].…”

“…It is easy to prove that our formula (3.2) is the same as equation (80) of Li et al [8]. Combination being made with the first order part of (2.33), namely 5) the solution (3.3) results in the first-order Eshelby tensor field which can be shown to be identical to that given by Mejak [14]. = α 0 7 − 5ν 10…”

“…Assume that the eccentric distance is h, the centre of the spherical inclusion can then be specified by h = hb(α, β) = h[e 3 cos β + (e 1 cos α + e 2 sin α) sin β], (1.2) where the origin of the coordinate system coincides with the centre of the spherical body. Adopting the notations introduced by Mejak [14], we define the dimensionless scalars The mth tensor power of x is denoted by x ⊗m , and its deviatoric (completely symmetric and traceless) part x ⊗p , or its components Ir(…”

“…The important Eshelby's problem of an inclusion embedded inside a three-dimensional (3D) finite body remains largely unsolved [1][2][3][4], even though the first attempts at solving it go back to the works of Kinoshita & Mura [5] and Kröner [6,7]. The study of Li and co-workers [8,9] seemed to initiate a new stage in dealing with this problem [10][11][12][13][14]. By using Somigliana's identity and Green's functions based on classical elasticity, Li and colleagues [8] first transformed the original problem into an integral equation one.…”

Analysis of a finite matrix with an inhomogeneous circular inclusion subjected to a non-uniform eigenstrain

“…Subsequently, based on the elasticity theory of a simplified strain gradient, Gao et.al [33] and Ma et.al [34] tested the similar problems and extended the solutions in the framework of high-order elasticity theory. Mejak [35] studied axisymmetrically placed inclusion in a sphere and found the Dirichlet-Eshelby tensor of an eccentric spherical inclusion through the power series expansion. Research on arbitrarily-shaped inclusion problems studied by Zou [36] is quite significant.…”