## 3 Classification of Core Topologies
### Core Topologies and Their Parameterisation  
New practicable, precise, and cost-effective manufacturing techniques [32; 33; 34; 35] led to a steadily increasing interest in innovative and efficient core geometries in recent years. This resulted in a large number of topology studies investigating the influence of geometric parameters on mechanical properties under various physical conditions and loading scenarios. However, hardly any attention is paid to macroscopic auxetic structuring. The focus here is, therefore, on classifying the performance of auxetic topologies in the broad performance spectrum of conventional non-auxetic core variants. The most common closed prismatic core topologies include square honeycomb cores and honeycomb cores composed of regular hexagonal cells. The behavior of these cores as a function of their relative density, the ratio of core height to cell size, the number of cells, and the impact of the use of different materials have been documented for a variety of impacts. For a vertical quasi-static compressive load, there is a linear relationship between the relative density, the maximum density, and the maximum compressive strength of the core construction. The influence of the cell size ratio is comparatively small [36]. The same applies to shear loading in the plane; the shear strength increases linearly with the relative density, while the cell wall dimensions play a subordinate role in relation to the core thickness [37]. The use of composite materials leads to qualitatively similar results. Here, the fiber direction of the carbon-fiber-reinforced polymer (CFRP) laminate decisively influences the system response [38; 39; 40]. With the help of a hierarchical core structure, which includes a sandwich construction for the cell walls, the buckling strength and, thus, the compressive strength can be significantly increased compared to conventional composite cores [41]. Dynamic compression tests with changing load application angles on aluminum honeycomb cores also proved that an increase in dynamic compressive strength and an almost invariable increase in shear strength take place with increasing impact velocity [42]. Figure 3 shows sections of the honeycomb cores examined in this contribution, including the associated unit cells and their respective topology parameters. In addition to the cell width \(\mathrm{b_{uc}}\) and the cell height \(\mathrm{h_{uc}}\), the inclination angle \(\alpha\) and the sheet thickness \(\mathrm{t_{c}}\) of the cell walls also determine the pattern geometry and mechanical behavior. The depth  
of the vertical prismatic cells results from the thickness \(\mathrm{h_{c}}\) of the core layer. In addition to the conventional honeycomb core, an isogrid variant (based on the regular honeycomb pattern), a quadratic core, and the standard auxetic honeycomb are considered. Open prismatic cores form a related class of core topology. The extrusion direction of the underlying unit cells is located parallel to the cover surfaces, resulting in core designs that are open on one side. The core geometries shown in Figure 4 are widely used in such cases; in principle, the same geometry parameters used for their closed counterparts can be used to adjust their mechanical properties. Extensive studies have already been conducted on this core group to determine their mechanical behavior in different loading situations. For example, triangular and diamond-shaped core designs have lower compressive and transverse shear stiffness than conventional honeycomb designs, regardless of the relative density. Only shear loads in the longitudinal direction can be absorbed in comparable orders of magnitude [43]. Structural optimization studies show that the weight efficiency of triangular and diamond-shaped core geometries is comparable to that of conventional honeycomb cores only at high compression load levels. Dependence on the load position is particularly pronounced for the triangular variant.