2.1. Composite Design Space, Fabrication, and Finite Element Analysis

Figure 1 shows the overall flow chart of this study. We aim to design and optimize the properties of a 2D composite composed of 16 × 16 soft and stiff cells; the optimization is performed on a fixed thickness. The material of each cell must be either soft or stiff, and the volume fraction of soft material is fixed at 12.5%. The boundary between the stiff and soft cells is strong and will not separate during the loading. The composite has an edge crack 25% of the specimen width oriented in the x-direction. The composite is symmetric about the x-axis (Figure 2). The resulting design space has possible designs of over 9.33 × 10¹⁹ combinations, which is unrealistic in terms of applying brute force algorithms to find optimal designs.

Overall flow chart.

Schematic of the composite design. The dimension of the design space is 13 mm × 13 mm. The displacement-controlled tensile test is conducted in the y-direction. The red box indicates the two elements behind the crack tip.

We chose two of the most important properties of load-bearing structures—stiffness and toughness—as our design targets. Stiffness is the slope of the linear part of the load-displacement curve, representing the composite’s ability to resist deformation under an applied load. Toughness is the area under the load-displacement curve representing the energy the composite can absorb before its crack propagates. According to the energy release rate failure criterion, a crack extends, i.e., fracturing, when the available energy release rate, G, for crack propagation is sufficient to overcome the material’s resistance to fracture G_c, i.e., G ≥ G_c [51]. The energy release rate, G, is the energy dissipation with a newly created fracture surface area and is defined as the rate of change in potential energy with a crack area. Thus, G may be regarded as the “driving force” for fracture; G_c is the fracture energy (critical energy release rate) and is a material property independent of the applied loads and the specimen geometry. In this study, since the material ahead of the crack tip is either soft or stiff (Figure 2), the corresponding G_c value is used for a particular design pattern. Accordingly, we introduced a fracture criterion, the ratio of the energy release rate and the critical energy release rate G/G_c, to compare the tendency of crack propagation among different patterns [52,53].

The FEM simulation was implemented by the finite element software ANSYS APDL 2020 R2 and linear hypothesis was used to gain the design targets, fracture criterion, and stiffness. The dimension of the simulation model was the same as the experiment specimens, as shown in Figure 3d. The designed pattern was in the middle of the specimen with a dimension of 13 mm × 13 mm and a 1/4 width length crack on one side edge in the middle. The element type was four-node plane182 with real constant, thickness t = 2 mm, and using mapped meshing, the element size at the design area was 0.1625 mm, and the mesh at the crack tip was refined fourfold to an element size of 0.040625 mm to represent the concentrated stress field better; the element size outside the design area is 1 mm. The upper and lower boundary displacements were pre-scribed in the y-direction with ±5 mm and constrained in the x-direction. Because the simulation of the energy release rate is quite simple and fast and can show how far it is for the crack to start propagating. Therefore, we assume that the fracture criterion can be used to predict the designed pattern’s toughness. Of course, the assumption must be validated by experiment.

We calculated the energy release rate by the strain energy difference before and after the crack extends 0.040625 mm, one crack tip element length. The formula is as follows [54]

where U₁ and U₂ are the strain energy before and after the crack extension, and da is the crack extending length (for two-dimensional problems). The units of G are J/m².

The stiffness is calculated by K = F/δ, where F is the sum of reaction forces of boundary nodes with applied displacement and δ is the difference of average displacements of upper and lower boundaries of the design area.

To validate the modeling results, we prepared the specimens using PolyJet multi-material 3D printing technique [55] (PolyJet, Stratasys). It works like an inkjet printer, but instead of jetting drops of ink, PolyJet 3D printers jet tiny droplets of liquid plastic on a substrate. A UV light instantly cures the plastic, solidifying it, so complex parts take shape layer by layer. This study’s stiff and soft materials were customized by mixing Vero (a stiff, white resin) and Agilus30 (a photosensitive PolyJet polymer black in color). The resultant stiff and soft materials were respectively FLX9870-70a with Young’s modulus E = 0.951 MPa and G_c = 249 J/m² and FLX9870-40a with Young’s modulus E = 0.455 MPa and G_c = 414 J/m². E and G_c are assumed isotropic within each cell, and G_c is for Mode I only. Those values for single materials were measured in this study using the same experimental setup to ensure consistent results with the composite ones. Our experiments show that PolyJet generally produces strong interface bonding between two different materials, whereas the commonly used extrusion-based 3D printing, i.e., FDM, produces a weak interface and can significantly affect the mechanical performance of multi-material printed structures [56]. Thus, FDM is unsuitable as its specimens almost always fail at the stiff-soft boundary instead of material fracture.