PSI - Issue 6

Giulio Zuccaro et al. / Procedia Structural Integrity 6 (2017) 236–243 Author name / Structural Integrity Procedia 00 (2017) 000–000

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Subsequently Mura (1982) coined the alternative terminology “eigenstrain” to represent a broad range of inelastic strains such as thermal strains, plastic strains, phase transformation, and misfit strains. According to Eshelby (1957) and Mura (1982), the subdomain containing the eigenstrain is called an inclusion to distinguish it from the term inhomogeneity which refers to a subdomain with a di ff erent material. As it is well known the Eshelby inclusion model and the related solutions represent the cornerstone of modern micromechanics thanks to which several advancements have been achieved in modern science and technology, Li and Wang (2008). To actually compute the elastic fields inside and outside the inclusion Eshelby evaluated the analytical expression of two potentials, namely an harmonic and a biharmonic one. This was possible only for inclusions of ellipsoidal (elliptical) shape though the expressions of the potentials were sensibly di ff erent inside and outside the inclusion. Although there have been extensive studies Zhou et al. (2013) related to the Eshelby inclusions, closed-form solu tions exist only for a limited number of cases, even for the two-dimensional (2D) inclusion in plane elasticity. Chiu (1980) obtained closed form solution of stress field in an elastic layer caused by a rectangular inclusion subjected to a uniform eigenstrain. Nozaki and Taya (2001) presented an exact solution to the stress field produced by a polygonal inclusion with uniform eigenstrain. Based on the complex variable method, Muskhelishvili (1953), Ru (1999) derived analytic solutions for an inclusion in a plane or half-plane while extensive references to the case of 3D inclusions can be found in Rodin (1996), Huang et al. (2011), Zhou et al. (2013), Trotta et al. (2017b). Two recent approaches to the evaluation of the Eshelby tensor and of the fields induced by a constant eigenstrain have been proposed in Trotta et al. (2016, 2017a) for the 2D case, and in Trotta et al. (2017b), for the 3D case. Remarkably all quantities of interest are obtained as algebraic sums whose scalar coe ffi cients depend upon the position vectors defining the polygonal (polyhedral) approximation of the inclusion boundary. In particular, displacements, strains, stresses and the Eshelby tensor are given by a unique expression inside and outside the inclusion, though recovering the correct jumps across its boundary, so that the derivation of distinct formu las for the elastic fields by di ff erent approaches exploited inside and outside the inclusion, can be completely avoided. This is in contrast with the previous contributions on the topic (see, e.g., Mura (1982), Ju and Sun (1999), Kim and Lee (2010), Jin et al. (2011), Jin et al. (2014), Jin et al. (2016) and Jin et al. (2017)). In particular, Jin et al. (2011) extended the analytical formulation developed in Ju and Sun (1999) and derived a novel expression of the Eshelby tensor correcting a previous one contributed in Kim and Lee (2010). The novel expression has been applied in Jin et al. (2014) to elegantly solve a classical problem in elasticity, namely stress concentration around elliptical holes in an infinite isotropic elastic plane, by the equivalent inclusion method. Subsequently, Jin et al. (2016) have developed explicit analytical solutions for displacements, displacement gra dients, strains and stresses, for both the interior and exterior fields, of an ellipsoidal inclusion. The specialization of their findings to plane strain has been reported in Jin et al. (2017) with reference to the elastic fields produced by a 2D elliptic cylindrical inclusion in a full plane with uniform eigenstrains. On the contrary we show that the same solution can be obtained by our formulation once the boundary of the elliptical inclusion has been approximated by a polygon having a number of sides, to be determined numerically, able to guarantee that the “numerical” Eshelby tensor does coincide with the exact one. Besides being more compact and recovering the classical solutions by Inglis (1913) and Maugis (1992), as in the quoted papers by Jin et al., the proposed approach can be successfully exploited to evaluate the elastic fields for problems in which the elliptical cavity is arbitrarily oriented with respect to non-uniform remote loadings. Let us consider an infinite, homogeneous and isotropic matrix containing an infinetely long cylindrical inclusion with uniform eigenstrain ε ∗ . We adopt a Cartesian reference frame of origin O and axis x 1 , x 2 orthogonal to the axis of the inclusion, see, e.g., Fig. 1 in Trotta et al. (2017a). We further denote by Ω the intersection of the cylindrical inclusion with the x 1 , x 2 plane and we assume that Ω has an arbitrary polygonal shape. The Eshelby tensor S is defined as the fourth-order tensor providing the strain tensor ε at an arbitrary poin P of the elastic medium through the relation ε ( ρ ) = S ε ∗ ( ρ ). In particular the Eshelby tensor is known to be constant if and only if the inclusion has an elliptical shape. Hence, under this assumption the elastic strain induced by ε ∗ is constant within the inclusion and depends upon the generic point within the host material (matrix) since the same happens for the Eshelby tensor 2. The displacement field and the Eshelby tensor for polygonal inclusions