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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assumed that materials are isotropic, they rarely are. There are two leading causes of anisotropy: one cause is the pre ferred orientations of grains or crystallographic texture; the second one is mechanical fibering. Natural materials like wood, rock, and biological tissues, as well as synthetic fiber-reinforced composites, can exhibit this microstructure feature at a significant level (Cowin, 2013; Hosford, 2010; Torquato and Haslach Jr, 2002; Dormieux et al., 2006). The importance of investigating the mechanical behavior of heterogeneous materials stems from several key rea sons, like material selection, predictive modeling, and failure prevention. In the micromechanics approach, the goal is to identify the mechanical response of heterogeneous materials from an understanding of their microstructure. The ef fective properties of two-phase materials are usually established assuming certain approximations of the actual stress and strain fields within a Representative Volume Element (RVE). Investigating these relationships helps uncover how microstructural features influence macroscopic properties, leading to insights into material behavior and potential for optimization (Dormieux et al., 2006; Qu and Cherkaoui, 2006; Suquet, 2014). In this context, the homogenization procedure is an important research tool for mathematically estimating the e ff ective macroscopic properties of heterogeneous materials based on the properties of the constituent phases and the microstructure geometry. Among homogenization techniques, the Mori-Tanaka (MT) method, derived from Eshelby (1957) is commonly used for poroelastic materials. Despite irregularly shaped pores being prevalent in heterogeneous materials, particularly those of natural origin, it is common practice to adopt ellipsoid pore shapes to assess the impact of irregular pores on mechanical properties. This approach stems from the ellipsoid’s ability to encompass a diverse range of irregular pore shapes. However, the issue of identifying the appropriate representative ellipsoid is still an open task. Several approaches have been proposed, such as moment of inertia classification and statistical modeling from micro-Computed Tomography ( µ -CT) analysis (Drach et al., 2013; Wang et al., 2018; Lee, 2018; Giraud et al., 2007). In the present study, we aimed to determine the e ff ective sti ff ness matrix of heterogeneous anisotropic materials based on representative ellipses of irregular pores obtained through the Mean Intercept Length (MIL) method com bined with Mori-Tanaka homogeneization model.

2. Methodology

This section describes the approach employed to investigate the e ff ective elastic properties in cases where the heterogeneous and oriented microstructure is taken into account. The aim is to present the theoretical framework that integrates the Mori-Tanaka homogeneization scheme with the Mean Intercept Length method. For the sake of clarity, the following notation is used henceforth: scalars are written as x , vector as x second-order tensors as X , and fourth-order tensors as X .

2.1. Microporomechanics

Microporomechanics involves scaling physical quantities from the microscale to the macroscale using a continuum approach. The analysis begins at the microscopic level, where the heterogeneous structure is explicitly represented through a Representative Volume Element (RVE). This RVE portrays distinct domains for the solid and void phases, as depicted in Fig. 1. The elementary volume is assumed as an infinitesimal part of the three-dimensional material system under consideration and is expected to be large enough to represent the constitutive material. More precisely, if we denote by L and ℓ the characteristic lengths of the structure and the RVE, respectively, the scale separability condition ℓ ≪ L ensures the use of continuum description tools (Dormieux et al., 2002; Ortega et al., 2010). Using this theoretical framework, the position vector z at the microscopic scale is employed to relate the micro scopic stresses σ ( z ) and strains ϵ ( z ) to their macroscopic counterparts Σ and E through volume averaging over the RVE (Ortega et al., 2010):

1 V V

σ ( z ) dV = σ ( z )

Σ =

(1)

1 V V

E

ϵ ( z )

= ϵ ( z )

=