# Improving the Impact Resistance of Anti-Ram Bollards Using Auxetic and Honeycomb Cellular Cores
## 2 Case Study and Modelling  
In this research, the bollard (Figure 1a) was modeled as a cylindrical shell that was 1500 mm in length, 76.2 mm in radius, and 8 mm thick, as also studied by [PERSON] et al. [49]. Modeling was conducted in Abaqus/CAE, while the analysis was performed using Abaqus/Explicit solver for hollow-pipe, honeycomb-core, and auxetic-core bollards. The cores (Figure 1b,c)were made of folded-welded sheets to create such a geometry. This method was adopted from [PERSON] et al. [35], and the dimensions were chosen to match bollard and core sizes. The geometrical parameters of the honeycomb and re-entrant auxetic cells were chosen based on a detailed parametric study [40] carried out earlier by the authors to find the most effective topologies that can result in higher energy absorption. As is known, a full 3D modeling of a detailed impacting car would lead to a heavy computational model with millions of degrees of freedom. Hence, a simple shell presenting the vehicle was used in the simulation as the research aims to improve the bollard's impact behavior rather than the car itself. The car body (Figure 1d) was modeled as a rigid body with a mass of 1500 kg (M1-class vehicles). Such an assumption is more conservative and would reveal a more critical impact on the bollard (conservative design). Regarding the material model, an elasto-plastic material model was used, which incorporates Johnson-Cook strain hardening and damage initiation. The Johnson-Cook material model is a constitutive model that replicates the plastic behavior of the material under high strain rates and high temperatures. In the Johnson-Cook model, the material flow stress \(\sigma\) considers strain rate hardening and thermal softening effects, as shown in Equation (1) [50; 51; 52; 53]:  
\[\sigma=\left(A+B\;\varepsilon^{n}\right)\,\left[1+C\ln\left(\frac{\dot{ \varepsilon}}{\dot{\varepsilon_{0}}}\right)\right]\,\left[1-\left(\hat{T} \right)^{m}\right] \tag{1}\]  
where, \(A\), \(B\), \(C\), \(n\), and \(m\) represent yield stress, pre-exponential factor, strain rate factor, work-hardening exponent, and the thermal-softening exponent, respectively. They are material parameters measured at or below room temperature. Moreover, \(\varepsilon\) is the plastic strain, \(\dot{\varepsilon}\) is the plastic strain rate, \(\dot{\varepsilon_{0}}\) is the reference plastic strain rate, and \(\hat{T}\) is the dimensionless temperature parameter that depends on the current material temperature, room temperature, and melting point of the material. Figure 1: Geometry and 3D views of the case study: (a) hollow-steel bollard; (b) honeycomb core; (c) auxetic core; (d) car body. The plastic strain at failure \(\varepsilon_{f}\) depends on non-dimensional plastic strain rate \(\frac{\dot{\varepsilon}_{\phi}}{\dot{\varepsilon}_{0}}\), pressure to Mises' stress ratio \(\frac{p}{q}\), dimensionless temperature parameter \(\hat{T}\) and (\(d_{1}-d_{5}\)) failure parameters, as shown below in Equation (2):  
\[\varepsilon_{f}=\left[d_{1}+d_{2}\text{exp}\bigg{(}d_{3}\ \frac{p}{q}\bigg{)}\right] \left[1+d_{4}\ \text{ln}\bigg{(}\frac{\dot{\varepsilon}}{\dot{\varepsilon}_{0}} \bigg{)}\right]\ \left(1+d_{5}\ \hat{T}\right) \tag{2}\]  
The material used for bollard pipe was Weldox 460E Steel [54] due to its high strength and ductility, as listed in [55]. Based on material mass density, in order to manufacture one hollow ballard shell, 46.2 kg of Weldox 460E Steel is required. On the other hand, the chosen material of the core (honeycomb and auxetic) was Aluminum Alloy AL7075-T651. The detailed material parameters/properties were adopted from [PERSON] et al. [56]. In order to manufacture one cellular core, 9.1 kg and 14.5 kg of AL7075-T651 are needed for honeycomb and auxetic cores, respectively. Hence, the total masses of the hollow bollard, honeycomb-core bollard, and auxetic-core bollard are 46.2 kg, 55.3 kg, and 60.7 kg, respectively.