## 2 Nonlinear Finite-Element Modeling
### Dynamic Explicit Solution Scheme  
The motion equation and its relevant initial conditions in explicit procedures can be expressed as follows:  
\[M\ddot{u}+C\dot{u}+Ku=P(t) \tag{1}\]  
\[u(t=0)=u_{0} \tag{2}\]  
\[\dot{u}=\frac{\partial u}{\partial t}|_{t=0}=v_{0} \tag{3}\]  
\[\ddot{u}=\frac{\partial u^{2}}{\partial^{2}t}|_{t=0}=a_{0} \tag{4}\]  
where \(M\), \(C\), and \(K\) are the mass, damping, and stiffness matrixes, respectively; \(t\) is the time; \(u\), \(\dot{u}\), and \(\ddot{u}\) denote the displacement, velocity, and acceleration, respectively; \(u_{0}\), \(v_{0}\), and \(a_{0}\) are the displacement, velocity, and acceleration at \(t=0\), respectively; and \(P(t)\) is the nodal forces vector.  
During the explicit procedures, the displacements are solved by using the central difference method, and the basic equation can be derived as:  
\[i\dot{u}|_{\left(t+\frac{\Delta t}{2}\right)}=i\dot{u}|_{\left(t-\frac{\Delta t }{2}\right)}+\frac{\left(\Delta t\right|_{\left(t+\Delta t\right)}+\Delta t \right|_{\left(t\right)})}{2}\ddot{u}|_{\left(t\right)} \tag{5}\]  
\[u|_{\left(t+\Delta t\right)}=u|_{\left(t\right)}+\Delta t|_{\left(t+\Delta t \right)}\dot{u}|_{\left(t+\frac{\Delta t}{2}\right)} \tag{6}\]  
where \(\Delta t\) is the time increment.  
The state of the model at time step \(t_{i}\) is directly derived from the previous step \(t_{i-1}\) so that an accurate result can be obtained if the increments are small enough. Nonetheless, the highly nonlinear behavior of LWAC results in variation of the highest frequency of the model, which would lead to a varying smallest stable temporal increment. Consequently, inconsistent time increments are requested. In fact, the central difference operator is conditionally stable, and the stability limit for the operator is given in terms of the highest of frequency of the system as (with no damping):  
\[\Delta t\leq\frac{2}{\omega_{\text{max}}} \tag{7}\]  
where \(\omega_{\text{max}}\) is the highest frequency of the system.  
With damping, the stable time increment is given by:  
\[\Delta t\leq\frac{2}{\omega_{\text{max}}}\left(\sqrt{1+\xi_{\text{max}}^{2}} -\xi_{\text{max}}\right) \tag{8}\]  
where \(\xi_{\text{max}}\) is the fraction of critical damping in the mode with the highest frequency.  
During the explicit procedure, an element-by-element method was used to estimate the stability limit for time increment, which is expressed as:  
\[\Delta t\approx\min\left(L_{e}\sqrt{\frac{\rho}{\lambda+2\dot{\mu}}}\right) \tag{9}\]  
where \(L_{e}\) is the element dimension, \(\rho\) is the density of the material in the element, \(\lambda\) and \(\dot{\mu}\) are effective [PERSON]'s constants for the material.  
Additionally, hereinto a small amount of damping is introduced in the form of bulk viscosity to control high-frequency oscillations, which is contrary to the usual engineering intuition. The bulkviscosity was introduced with truncation frequency damping. It generates a bulk viscosity pressure \(P_{bv1}\) that is linear in the volumetric strain rate:  
\[p_{bv1}=b_{1}\rho c_{d}L_{e}\varepsilon_{vol} \tag{10}\]  
where \(b_{1}\) is the damping coefficient of 0.06, \(c_{d}\) is the current dilatational wave speed, and \(\varepsilon_{vol}\) is the volumetric strain rate.