## 2 Materials and Methods
### Analytical Method  
An inhomogeneous solid in which the inhomogeneities are distributed periodically can be considered as a lattice structure. In artificial lattice structures, such periodic patterns are used to help reduce the amounts of materials used compared to making original solids while retaining the desired properties of the original solids. If a lattice structure consists of a substantial number of unit cells, then the effective properties of the structure can be determined [45]. An infill specimen in this study is constructed by a lattice structure and bounded by side walls. Since the infill pattern of the MAM parts is periodic, the infill specimens with the triangular pattern can be treated as lattice structures with triangular unit cells. Figure 5 shows the lattice structure with triangular unit cells, in which \(l\) denotes the characteristic length of the unit cells while \(b\times t\) denotes cross-sections of the unit-cell struts. The theoretical relative density of a lattice structure with triangular unit cells can be expressed as [46]  
\[\overline{p}_{tri}=2\sqrt{3}\frac{b}{l} \tag{1}\]  
where \(b\) and \(t\) denote the width and thickness, respectively; \(b\) is designed to be a constant, \(l\) can be varied to adjust the relative density of a lattice structure. The elastic properties of the lattice structures with the triangular unit cells can be found in the literature. Analytical forms of the effective elastic modulus \(E_{tri}\) and yield strength \(\sigma_{y,\ tri}\) of the lattice structure with the triangular pattern derived by [PERSON] and [PERSON] [17] can be written as  
\[E_{tri}=0.333\overline{p}_{tri}E_{s} \tag{2}\]  
Figure 5: A lattice structure with triangular unit cells. \[\sigma_{y,\ tri}=0.5\overline{p}_{tri}\sigma_{ys} \tag{3}\]  
where \(E_{s}\) and \(\sigma_{ys}\) denote the elastic modulus and yield strength of the base material, respectively. Equations (2) and (3) were constructed with the assumption that the struts were considered as short columns with \(b/l\approx 0.1\) to 0.15. For Equation (3), the perfectly plastic deformation was considered for every strut. The effective yield strength was based on the first strut that achieved the yield limit. Moreover, in a study by [PERSON] et al. [19], a generic symbolic finite element program was used to derive the closed-form effective elastic modulus of lattice structures with triangular unit cells using the strain-energy-based homogenization method. The struts of the lattice structure were modeled as frame structures using Euler's beams. The effective elastic modulus of the lattice structure with triangular unit cells is written as  
\[E_{tri}=\frac{l^{2}+b^{2}}{3l^{2}+b^{2}}\overline{p}_{tri}E_{s} \tag{4}\]  
Since the lattice structure with triangular unit cells is bounded by the two side walls, as shown in Figure 6, it is necessary to include the stiffness of the side walls into the overall properties. Rule-of-mixtures models or strength-of-materials models [47] were used to determine the overall elastic modulus \(E_{overall}^{*}\), i.e.,  
\[E_{overall}^{*}=\frac{W_{tri}}{W}E_{tri}+\frac{W_{s}}{W}E_{s} \tag{5}\]  
where \(W_{tri}\) and \(W_{s}\) denote the widths of the lattice structure and the solid wall portion, respectively. In addition, \(W\) denotes the overall width of the specimen. Similarly, the overall yield strength can be derived using the rule-of-mixtures model. The overall yield strength \(\sigma_{y,overall}^{*}\) divided into two conditions, depending on which portion reaches the yield point first, can be expressed as  
\[\sigma_{y,overall}^{*}=\frac{W_{tri}}{W}\sigma_{y,\ tri}+\frac{W_{s}}{W}\sigma _{s}^{\prime} \tag{6}\]  
Figure 6: The overall structure of a specimen and its components: solid wall portion (wall side thickness) and triangular lattice portion (interior).