# Impact Testing of 3D Re-Entrant Honeycomb Polyamide Structure Using Split Hopkinson Pressure Bar
## 2 Related Theories on the SHPB Test  
SHPB technology is widely used to carry out impact tests on specimens to obtain the dynamic properties of materials. SHPB is usually made of high-strength steel; however, the specimens in this study are of a 3D RH polyamide structure, and the mechanical impedance and strength of the specimens are much lower than that of steel bar. If the SHPB steel bar with high impedance acts on the specimen with low impedance, the incident waveform and reflected waveform will be very close, and the transmissive wave signal will be too weak to be detected accurately. Therefore, an SHPB bar whose impedance is close to the specimen should be used [29; 30; 31]. In this study, the viscoelastic polymethyl methacrylate (PMMA) bar was used, and the mechanical impedance of the PMMA bar was relatively matching with the specimen. It should be noted that the wave attenuation caused by material damping and the wave dispersion effect caused by radial inertia should be considered on the wave propagation of the viscoelastic bar. [PERSON] [29] developed an experimental method to evaluate the propagation coefficient. [PERSON] and [PERSON] [32] presented an analytical solution of the longitudinal wave propagation in a viscoelastic cylindrical bar. [PERSON] et al. [31; 33] developed a method based on the Zhu-Wang-Tang (ZWT) viscoelastic constitutive equation and the characteristic theory of wave propagation. According to some studies [34], if the minimum wavelength of the impact wave is significantly greater than the lateral dimensions of the bar, the dispersion effect caused by radial inertia can be neglected.  
For wave attenuation caused by material damping, the waveform at different locations varies with distance within a limited propagation distance (up to 3 m in bar length). Therefore, the waveform measured by the test at the position of strain gauge on the SHPB bar cannot directly replace the waveform at the interface between the bar and the specimen. In this study, an improved Lagrange analysis method [35], which was verified in an earlier study, was used to process the strain data measured on the SHPB bar, and then the incident strain wave \(\varepsilon_{I}(X_{1},t)\) and transmission strain wave \(\varepsilon_{T}(X_{2},t)\) at the two interface sites could be obtained, respectively. The wave dispersion effect caused by radial inertia was neglected, and the ZWT viscoelastic constitutive equation and characteristic method were used to solve the strain, stress and velocity of all positions on the PMMA bar. These data were compared with the data obtained by the improved Lagrange analysis method to verify the validity of dynamic data processing on the PMMA bar. It is worth noting that this paper mainly studies the impact behaviour of the 3D RH structure, rather than the dynamic testing process and data processing of the PMMA bar. The corresponding content will be detailed in other papers published by the authors.  
Based on the one-dimensional stress wave hypothesis, the stress and strain of the SHPB impact specimen can be calculated by using the one-dimensional stress wave theory, which can be expressed as:  
\[\sigma_{s}(t)=\frac{EA}{A_{s}}[\varepsilon_{I}(X_{1},t)+\varepsilon_{R}(X_{1},t )]=\frac{EA}{A_{s}}\varepsilon_{T}(X_{2},t), \tag{1}\]  
\[\varepsilon_{s}(t)=\frac{2C_{0}}{l_{s}}\int_{0}^{t}\varepsilon_{R}(X_{1},t)dt= \frac{2C_{0}}{l_{s}}\int_{0}^{t}[\varepsilon_{T}(X_{2},t)-\varepsilon_{I}(X_{1 },t)]dt, \tag{2}\]  
where \(\sigma_{s}(t)\) and \(\varepsilon_{s}(t)\) are the average stress and strain of the specimen, respectively. \(A_{s},l_{s}\) represent the cross-section area and length of the specimen, respectively. \(E,A,C_{0}\) represent the Young's modulus, the cross-section area and the longitudinal elastic wave velocity of bar, respectively.