PSI - Issue 61

Lívia Mendonça Nogueira et al. / Procedia Structural Integrity 61 (2024) 122–129 L. M. Nogueira et al. / Structural Integrity Procedia 00 (2024) 000–000

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Fig. 1. Heterogeneous material microstructure with microscopic RVE and macroscale homogenization.

In the case of microporoelasticity, the microscopic strain tensor ϵ ( z ) depends linearly on the components of the macroscopic strain tensor E by means of a fourth-order strain concentration tensor, denoted by A ( z ), as established byEq. 2. ϵ ( z ) = A ( z ) E (2) In elastostatics, the computation of the strain concentration tensor is traditionally attributed to Eshelby’s research Eshelby (1957) who formulated the elastic field for an ellipsoidal inclusion embedded in an infinitely large, homoge neous, isotropic, linear elastic solid matrix. The ellipsoidal schematic depiction is illustrated in Fig. 2, where the local coordinate system attached to the ellipsoid inclusion is represented by directions I , II , and III .

Fig. 2. Ellipsoidal inclusion domain embedded in the matrix.

Eshelby investigated inclusions in materials, discovering uniform strain and stress fields within them, as described in Eq. 3 (Eshelby, 1957; Meng et al., 2012; Parnell, 2016): ϵ i j = S i jkl E kl (3) where S i jkl are components of the Eshelby tensor S . By comprehending the manner in which stress and strain propagate from inclusions to the surrounding medium, this information is integrated into homogenization techniques. In this regard, the Mori-Tanaka scheme is considered, as it models pores as individual inclusions within a solid matrix, making it suitable for situations with low porosity, usually below 40% (Benveniste, 1987; Dormieux et al., 2006; Torquato and Haslach Jr, 2002; Dunant et al., 2013). The e ff ective sti ff ness tensor for fully drained conditions, following the Mori-Tanaka homogenization approach, can be expressed in local coordinates. These local coordinates are aligned with the reference inclusion, and the tensor can be written as: C MT = (1 − f p ) C m : (1 − f p ) I + f p ( I − D : C m ) − 1 − 1 (4)