## 2 Synergistic Mechanical Design of Strength and Toughness with Ceramic-Infilled Re-Entrant Lattice  
When materials and structures with a negative Poisson's ratio are compressed, they will shrink laterally. In the following, the classic re-entrant auxetic lattice structure is referred for design and fabrication. As shown in Figure 1a, \(l\) is the width of honeycomb cells, \(h\) is the height of honeycomb cells, \(\theta\) is the concave angle, t is the honeycomb cell wall thickness, and in the following examples, the geometrical parameter \(l=h\) is specified. The main reason for the formation of a negative Poisson's ratio within re-entrant honeycomb material is axial compression and hinge rotation, which makes the inclined rib bend inward, resulting in simultaneous contraction of the structure in both transverse and longitudinal directions, as shown in Figure 1b. If the ceramic is filled into re-entrant hexagonal honeycomb frames, the built-in ceramic will be subject to a lateral constraint so that the hybrid re-entrant auxetic lattice with built-in ceramic can be employed for improving both strength and toughness. As shown in Figure 1c,f, design and mechanical tests of the unit cell consisting of different types of hexagonal honeycomb structures are performed first. When the hexagonal unit cell is compressed, internal defects (cracks, voids, etc.) will appear along the horizontal direction, as shown in Figure 1d. Ideally, when the negative Poisson's ratio unit cell is compressed along vertical direction, cracks will not spread due to metal frame contractions along both the vertical and horizontal directions, as shown in Figure 1g. In order to verify the influence of the external metal frame on the built-in ceramic matrix, this paper verifies the stress state of ceramics within this hybrid material unit cell under compression loading conditions from the perspectives of simulation and experiment comparisons. The internal cell width and internal cell height of the metal honeycomb structure are \(l=h=35\) mm, the thickness t = 5 mm, and the depth of the structure is 10 mm. Figure 11,j illustrate the sideview of quasi-static compression finite element analysis results of single cell hybrid materials, in which only the horizontal stress contour plot of the built-in ceramic is shown. Conventional and re-entrant hexagonal metal ceramic composite structures are subject to vertical compression at a loading speed of 0.004 mm/s. In the numerical calculation of quasi-static compression, the constitutive model of the outer metal  
Figure 1: Re-entrant hexagonal auxetic lattice (**a**) before compression (**b**) after compression; single unit cell metal-ceramic hybrid materials (**c**–**e**) conventional and (**f**–**h**) re-entrant hexagonal honeycomb; transverse stress distribution of ceramic materials while the hybrid material is subjected to longitudinal pressure (**i**) at the beginning of compression (at strain value 0.5%), (**j**) at strain value 8%. frame used in the FEM model is the Johnson-Cook model [28] and the constitutive model of inner ceramic in the FEM model is the Drucker-Prager Cap model [29]. The FEM model of built-in ceramic was simulated with brick (C3D8R) elements with max Warpage = 0, max aspect ratio = 5.44, max Skew = 0, min Jacobian = 0.712, min angle = 60\({}^{\circ}\) and max Angle = 60\({}^{\circ}\). Figure 1i,j also shows the contour of horizontal stress within ceramic materials. At the beginning of compression, for the ceramic in the ordinary honeycomb, 73.3% of the volume is transversely compressed and the average horizontal stress is \(-\)12.8 MPa, and this value is 85.4% for the transversely compressed re-entrant honeycomb. The average horizontal stress is \(-\)108.3 MPa. For the ceramic within the ordinary honeycomb, the horizontal stress of most ceramic elements is between (\(-\)100, +27) MPa. However, for the ceramic in the re-entrant honeycomb, the horizontal stress of the ceramic phase is relatively evenly distributed between (\(-\)450, +27) MPa.