# Mechanical Mechanism and Dynamic Characteristics of Barge-Whole Bridge Collision Behaviours
## 2 Materials and Methods  
### Material Constitutive Models and Failure Criterion  
In the process of a ship impacting a bridge, the concrete pier of the bridge mainly undergoes the extruding force arising from the ship. The Holmquist-Johnson-Cook (HJC) constitutive model [15] properly express the constitutive law of concrete materials subjected to large strains, high strain rates and high pressures, with the advantages of comparatively precise simulation, convenient calibration of material parameters, and high calculation efficiency. It is capable of taking the compression effect, strain rate effect and damage softening effect into account, and it is regarded as one of the most reasonable constitutive models for characterizing collision-induced damage problems in concrete structures. The yield surface equation and damage evolution equation of the HJC constitutive model [16; 17] are the followings:  
\[\sigma^{*}=\frac{\sigma}{f_{c}^{\prime}} \tag{1}\]  
where \(f_{c}^{\prime}\) is the quasi-static uniaxial compressive strength; \(\sigma\) is the actual equivalent stress; and \(\sigma^{*}\) is the normalized equivalent stress, defined by [16; 17]:  
\[\sigma^{*}=\left[A(1-D)+B\left(P^{*}\right)^{N}\right]\left(1+C\dot{n}\dot{ \varepsilon}^{*}\right) \tag{2}\]  
where, \(P^{*}=P/f_{c}^{\prime}\) refers to the normalized pressure, with \(P=(\sigma_{1}+\sigma_{2}+\sigma_{3})/3\), the hydrostatic stress tensor; \(\dot{\varepsilon}^{*}=\dot{\varepsilon}/\dot{\varepsilon}_{0}\), the normalized strain rate, with \(\dot{\varepsilon}_{0}\) being the reference strain rate; \(A\), normalized cohesion strength; \(B\), normalized pressure hardness coefficient; \(N\), hydrostatic pressure hardening coefficient; \(C\), strain rate variation coefficient; the damage coefficient, \(D\), considering equivalent plastic strain and plastic volumetric strain, expressed by [16; 17]:  
\[D=\sum\frac{\Delta\varepsilon_{p}+\Delta\mu_{p}}{\varepsilon_{p}^{\prime}+\mu _{p}^{\prime}} \tag{3}\]  
\[\varepsilon_{p}^{f}+\mu_{p}^{f}=D_{1}(P^{*}+T^{*})^{\Omega_{2}}\geq EMIN \tag{4}\]  
where \(\Delta\varepsilon_{p}\) and \(\Delta\mu_{p}\) are the equivalent plastic strain and plastic volumetric strain in each cycle of integration, respectively; \(\varepsilon_{p}^{f}+\mu_{p}^{f}\) is the plastic strain to fracture under constant pressure, \(P\); \(D_{1}\) and \(D_{2}\) are both damage constants; \(T^{*}=T/f_{c}^{\prime}\), the maximum normalized tensile force; \(EFMIN\), minimum of plastic strain before fracture. The diagrams of yield surface and the damage evolution are plotted in Figure 1[16; 17] with the relevant material parameters presented in Table 1.  
Figure 1: Description of HJC constitutive model: (**a**) representation of yield surfaces; (**b**) visualization of damage evolution function [16; 17]. In the FE collision simulation, the elements of the pier are distorted due to the impact, which does not conform to the reality [18,19]. With the HJC constitutive model, the concrete failure criterions were added in this study. The maximum pressure of 35 MPa and the maximum principal strain of 0.02 were set as the failure criteria of concrete for obtaining more realistic damage effects. The actual simulation indicates that adding failure criterion can provoke more reasonable dynamic responses and characterization of damage evolution process of the barge impacting the bridge. The material of the barge bow was characterized by the plastic kinematic model [20]. This model can describe the behavior of isotropic hardening and the kinematic hardening plastic model, as well as the effect of strain rate. Without considering the influence of strain rate, the stress-strain relationship of the model is shown below:  
\[\sigma_{y}=\sigma_{0}+\beta E_{P}e_{eff}^{P}, \tag{5}\]  
where \(\sigma_{y}\) is the yield strength; \(\sigma_{0}\) is the initial yield strength; \(E_{P}\) is the plastic hardening modulus; \(\beta\), hardening parameter; and \(e_{eff}^{P}\) is the effective plastic strain.