## 2 Methodology
### Design of Unit Cells and Lattice Structures  
In this study, FGLSs with different configurations were designed using auxetic unit cells with advantages such as high energy absorption and impact resistance. Modified re-entrant star honeycomb, modified chiral and modified rotation unit cells were parametrically and knowledge-based modelled with dimensions of 5 \(\times\) 5 \(\times\) 5 mm in grasshopper. Two important features for the first of these unit cells, the re-entrant star honeycomb, are the re-entrant parts and the corners with hinge function. In the literature, the mechanism of this unit has been analysed assuming that the re-entrant parts and corners are elastically supported hinges where each element of the structure is modelled as a strut [43; 44]. Star-shaped lattice structures have been extensively studied due to their attractive mechanical and physical properties such as negative Poisson's ratio, low weight, high strength and high energy absorption. The star honeycomb unit cell chosen for this study was a simple model consisting of four straight beams of equal length (\(L1\)) joined by four re-entrant corners of equal length (\(L2\)) [45]. The corner angle between adjacent cell walls is denoted by \(\alpha\) and the lattice constant is denoted by \(W\) (Figure 1c). [PERSON] et al. (2018) reported the relationship between geometric parameters for the re-entrant star honeycomb unit cell (Equation (1)) [45]:  
\[W=2\times\left\{\frac{\sin(\alpha-45^{\circ})}{\sin(45^{\circ})}\times L2+L1\right\} \tag{1}\]  
The parameter settings for the re-entrant star honeycomb model: straight beam length \(L1=1.67\) mm, re-entrant corner length \(L2=2.27\) mm, corner angle \(\alpha=60^{\circ}\) and lattice constant \(W=5\) mm. The re-entrant star honeycomb model, which is mostly used as a two-dimensional (2D) model in the literature, was converted into 3D by multiplying it in different axes and used as a unit cell in the design of lattice structures in this study. Another cellular unit used in this study is the 3D chiral structure. [PERSON] et al. (2022) proposed a new 3D chiral metamaterial that allows modular design [46]. The proposed chiral cellular unit consists of six independent units, useful for tailoring anisotropic chiral properties and combining unit cells with different geometries. In this study, for the 3D chiral unit, the prism length of the inner cube a, the edge length of the outer frame (b) and the outer edge length of the unit cell (Lc) are equalized, and the beams of the outer frame are extracted (Figure 1a). The last unit cell used in this study was modified rotating squares. The rotating squares mechanism is one of the leading robust, feasible auxetic models [47]. The rotating square's structure can easily turn corners when tensile force is applied. Due to its geometry, it can expand both upwards and downwards when under load [48; 49]. [PERSON] and [PERSON] (2000) presented a detailed analysis of the relationships between rigid squares connected by hinges at the corners [48]. They expressed the size of the auxetic structure for a set of four connected squares, as shown in Equation (2). The auxetic rotating unit cell, which is mostly used in 2D in the literature, was converted to 3D in this study and used to form a lattice structure. The dimensions of \(X1\) and \(X2\) are both equal to 5 mm and the angle \(\theta\) between the beams is 30\({}^{\circ}\) (Figure 1b). \[X1=X2=2l\left(cos\frac{\theta}{2}+sin\frac{\theta}{2}\right) \tag{2}\]  
The \"intralattice\" plugin of the grasshopper software was used for the design of the lattice structures in order to allow the unit cells to increase periodically. In this plugin, the customized unit cell designs created within the scope of this study have been added. FGLSs with uniform, vertically graded (VG), horizontally graded (HG) and radially graded (RG) configurations were created for each auxetic unit cell geometry (Figure 2). The strut diameters of the cellular structures have a gradient structure ranging from 0.4 to 1 mm.