*2.3. Basic Cell*

In the research unit, a cube with a spheroidal pore was used as a basic cell. In this case, the parameters vector consists of porosity (*λ*) and the ellipticity coefficient (*β*). To investigate the dependence between the stiffness tensor and the parameters vector, a representative elements method was used [31–33]. For this purpose, the parameterized finite element model of a cube with a spheroidal pore was implemented. Twenty-node hexahedral finite elements were used. Kinematic loading was used in the numerical simulation. Uniaxial and shear loads in three directions were implemented. To clarify the mechanical properties, additionally combined (uniaxial with shear) loads were implemented [16,26,34,35]. The parameters were investigated in the interval (0; 1). According to the received data, the functions describing the influence of the parameters on the mechanical properties were found:

$$\begin{cases} \stackrel{\rightarrow}{\lambda} = \stackrel{\rightarrow}{\lambda}(\lambda, \beta) \\ E\_{ii} = E\_{ii} \left( \stackrel{\rightarrow}{\lambda} \right) \equiv \mathbb{C}\_{iiii} \left( \stackrel{\rightarrow}{\lambda} \right) \\ \mathcal{G}\_{ij} = \mathcal{G}\_{ij} \left( \stackrel{\rightarrow}{\lambda} \right) \equiv \mathbb{C}\_{ijij} \left( \stackrel{\rightarrow}{\lambda} \right) \\ \nu\_{ij} = \nu\_{ij} \left( \stackrel{\rightarrow}{\lambda} \right) \equiv \mathbb{C}\_{iiij} \left( \stackrel{\rightarrow}{\lambda} \right) \end{cases} \tag{12}$$

For approximation, a fourth-degree polynomial function was used with an approximation error threshold of about 0.9 [33–36]. In the calculations, some of the coefficients were equal to zero, so a common form of the final calculated polynomial was as follows:

$$\mathcal{C}\_{ijkl}(\lambda\_\nu \mathfrak{f}) = \mathfrak{c}\_{00} + \mathfrak{c}\_{10}\lambda + \mathfrak{c}\_{01}\mathfrak{f} + \mathfrak{c}\_{11}\lambda \mathfrak{f} + \mathfrak{c}\_{21}\lambda^2 \mathfrak{f} + \mathfrak{c}\_{31}\lambda^3 \mathfrak{f} + \mathfrak{c}\_{12}\lambda \mathfrak{f}^2 + \mathfrak{c}\_{22}\lambda^2 \mathfrak{f}^2 + \mathfrak{c}\_{13}\lambda \mathfrak{f}^3 \tag{13}$$

where *λ* and *β* are components of the parameters vector—porosity and ellipticity, respectively, *cij* are coefficients of the polynomial, where *i* shows the power of porosity and *j* shows the power of ellipticity. The received values of the coefficients for the approximation polynomial are listed in Table 1.


**Table 1.** The values of coefficients of approximation polynomial for stiffness parameters.

It should be noted that the polynomial coefficients for Poisson's ratio can be reduced up to *c*<sup>00</sup> because the influence of the parameters vector is insignificant. So, *ν*12,13 ≈ 0.011 and *ν*<sup>23</sup> ≈ 0.017.
