*4.2. Dynamic Wave Verification*

To validate the effectiveness of the developed simplified Johnson–Cook model of FFRCs, numerical simulation was conducted using the commercial FEM software ABAQUS 6.13 (Dassault Systemes S.A, Vélizy-Villacoublay, France) to simulate the SHPB experiment.

A three-dimensional FEM model was set up to simulate the SHPB experiments. The FEM model consisted of four components: a striker bar, an incident bar, a transmission bar, and an FFRC specimen, each of which was of the same size as the SHPB apparatus shown in Figure 3. To improve the accuracy of the FEM analysis, an 8-node linear brick with reduced integration and hourglass control (C3D8R) was adopted. The minimum size of all elements was 1 mm. The same material was used for the three components made of steel, with the following measurements: modulus, 190 Gpa; density, 8 g/cm3, and Poisson ratio, 0.3. The developed simplified Johnson–Cook model was used to simulate the mechanical behavior of the FFRC specimen in Table 2. The time history of the strain wave was obtained from the same location of the strain gauges on the incident and transmission bars with SHPB experiments.

**Table 2.** Fitting parameters in the simplified Johnson–Cook model.


The speed of the striker was varied to obtain the various loading conditions of the tested specimens. The strain wave obtained from the experiments, with its counterpart from the FEM calculations, is shown in Figure 8. Consistency was found, indicating the validity and accuracy of the simplified Johnson–Cook material model. However, a slight general difference still observed, which could be attributed to the following: (a) The influence of thermal softening under impact could not be simulated in ABAQUS Explicit [25]; (2) The geometry of the specimens was not perfectly cubic [37]; (3) The non-parallelism and friction between the faces in contact with the bars were hardly included in the FEM model [37]; (4) The strain rate was not constant in the SHPB, and so on.

**Figure 8.** Comparison of strain waves between the simplified Johnson–Cook model and experimental data under strain rates (**a**) 1300 s<sup>−</sup>1, (**b**) 2200 s<sup>−</sup>1.

### *4.3. Prediction of the Crushing Peofrmance of Lattice Structures*

As in Section 4.2, to further validate the effectiveness of the developed simplified Johnson–Cook model of the FFRC applied in engineering structure analysis, numerical simulation was also conducted using ABAQUS to simulate lattice structures subjected to quasi-static crushing and drop-hammer impact.

### 4.3.1. Specimen and FEM Model

Lattice structures were man-made open, porous cellular solids with periodic truss microstructures [38,39], which could meet many stringent requirements of engineering applications, such as blast and ballistic resistance, impact load carrying, and energy absorption. In the present study, FFRCs were used to manufacture pyramidal lattice cores with additional horizontal trusses for structure crashworthiness applications. The manufacturing process is illustrated in Figure 9. Six types of truss strips were cut with a carving machine (3040, Shenzhen Yidiao, Shenzhen, China) with a cutting precision of 0.03 mm from an FFRC sheet and then assembled into a pyramidal lattice structure by strip slot insertion (see Figure 9a–d). To fix the lattice core, two pieces of glass fiber-reinforced composite panels (see Figure 9e) were used to bond both sides of the lattice core forming a sandwich structure (see Figure 9f). The specimen of the lattice structure measured 84 mm in length, 84 mm in width, and 14 mm in height.

**Figure 9.** Pyramid lattice manufacturing: (**a**) Cutting truss strips of the lattice core from the fabricated FFRC; (**b**) obtaining six types of truss; (**c**) assembling truss into a pyramidal lattice structure by strip slot insertion; (**d**) assembled lattice structure; (**e**) top view of the specimen; (**f**) left view of the specimen; (**g**) FEM model of the specimen.

Numerical models were then established to predict the crushing performance of the lattice structure under quasi-static and drop-hammer impact conditions in the out-of-plane loading direction. A three-dimensional FEM model with the C3D8R element type was numerically established in ABAQUS, (Figure 9g). The density of the elements was relatively higher in the core, and the element size was 0.3 mm. The FEM model had exactly the same size as the tested specimen with setups. Two plates were considered as rigid bodies to crush the specimen.
