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The time discretization scheme may be selected in a manner that enforces linear and angular momentum, and energy conservation over consecutive time steps $t_{n}$ and $t_{n+1}$ , when boundary conditions and external loads are time-independent. Based on the prior literature [69, 34, 62], this momentum and energy conservation may be achieved by using the midpoint rule ($\rho_{\infty}=1$ , leading to $\alpha_{f}=\alpha_{m}=\frac{1}{2}$ ), and evaluating the virtual work at $t_{n+\frac{1}{2}}$ . However, since the virtual work strictly enforces momentum balance only, there is no guarantee that energy conservation will be satisfied as a result of time discretization. Therefore, we need to enforce a specific scheme to satisfy energy balance.